RESEARCH STATEMENT SUSANNA DANN

1. Introduction My current research interests lie in the areas of harmonic analysis and convex geometry. The starting point of my research and a motivation to pursue graduate studies was image processing, where I worked on evaluation of image data for automotive manufacturing and medical imaging. Abstract harmonic analysis was the subject of my graduate research. During my postdoctoral studies, I applied harmonic analysis to problems in convex geometry. One of my latest projects is an exciting collaboration with researchers working on probabilistic aspects of convex geometry. The beginnings of convex geometry date back to ancient Greek mathematics. Today, it is an active field naturally related to many areas of mathematics and science such as functional analysis, probability theory, linear programming and information theory. In recent years, techniques of harmonic analysis have been extensively used to solve questions in convex geometry. Likewise, efforts to solve problems in convex geometry gave rise to now very familiar harmonic analysis tools, like the Radon transform [8]. One of my most recent and ongoing projects is related to affine isoperimetric inequalities. Let me start with probably one of the best known results, not only in convex geometry but in mathematics at large, with the isoperimetric inequality in the plane. It asserts that for any closed curve of length L enclosing a planar region of area A, we have 4πA ≤ L2 , and that equality holds if and only if the curve is a circle. The area A is of course invariant under all linear volume preserving transformations, while the length L is invariant only under rotations. In convex geometry a big class of inequalities is termed ‘isoperimetric’ [29]. For instance the following inequality due to H. Busemann is isoperimetric. Let K be a convex body in Rn , then Z n ωn−1 ⊥ n (1.1) Voln−1 (K ∩ θ ) dσ(θ) ≤ n−1 Voln (K)n−1 . ωn Sn−1 Here Volk (·) stands for the k-dimensional Euclidean volume, ωk stands for the volume of the k-dimensional Euclidean unit ball, σ is an invariant probability measure on Sn−1 and θ⊥ stands for the hyperplane perpendicular to θ. Equality holds in (1.1) if and only if K is an ellipsoid. Both sides of (1.1) are invariant under linear volume preserving transformations, as was shown by E. Grinberg [18]. Such an invariance property of a geometric quantity is very valuable since most of them cannot be computed explicitly, even for simple objects such as Date: October, 2017. 1

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the L1 -unit ball. Furthermore, it gives an insight into the behavior of the geometric object in question, for example about its extremizers. In [13] we found powerful generalizations of (1.1) and other related geometric inequalities, from sets to bounded integrable functions and proved their (linear) affine invariance properties. Moreover, we established a link between these geometric inequalities and purely probabilistic statements about marginal densities of a bounded probability distribution. This project is described in 2.1 below. In [11] we study (1.1) in the real hyperbolic and spherical spaces, see 3.5 for a brief description. Another principal branch of convex geometry is the study of geometric properties of convex bodies based on the information about sections and projections of these bodies. For example, the famous Busemann-Petty problem asks the following question. Given two origin-symmetric convex bodies K and L in Rn such that Voln−1 (K ∩ H) ≤ Voln−1 (L ∩ H) for every hyperplane H in Rn containing the origin, does it follow that Voln (K) ≤ Voln (L)? The answer to this problem is affirmative for n ≤ 4 and negative for n ≥ 5. This question was posed in 1956 and solved in the late 90’s as a result of a sequence of papers. The analytic solution of this problem is based on Fourier techniques. My contributions to this branch of convex geometry include the solution of the Busemann-Petty problem in the complex hyperbolic space [9], as well as the lower-dimensional version of this problem in the same space [10]; these are outlined in 3.1 and 3.2. A class of bodies, called intersection bodies, played an important role in the solution of the Busemann-Petty problem in Rn . One of the main steps in the solution was the connection established by E. Lutwak in [28] between this problem and intersection bodies. It was found that for an intersection body K and any star body L, the Busemann-Petty problem has a positive answer. Yet, for an origin-symmetric convex body L that is not an intersection body, one can construct a counterexample. Recall that a compact subset K of Rn containing the origin as an interior point is called a star body if every line through the origin crosses the boundary in exactly two points different from the origin. Intersection bodies in Rn have been a subject of many studies. However, their geometric properties remain a mystery. In [14], we generalize the class of intersection bodies in Rn by imposing invariance under a certain subgroup of orthogonal transformations and show that this class of bodies shares many properties with the original class of intersection bodies in Rn . This project is sketched in 3.3. The average section functional as(K) of a centered convex body in Rn is the average volume of the central hyperplane sections of K: Z as(K) = Voln−1 (K ∩ ξ ⊥ ) dσ(ξ). S n−1

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In [4] we study the question whether there exists an absolute constant C > 0 such that for every n, for every centered convex body K in Rn and for every 1 6 k 6 n − 1, k

as(K) 6 C k Voln (K) n

max

E∈Grn,n−k

as(K ∩ E),

where Grn,k stands for the Grassmann manifold of all k-dimensional subspaces of Rn . We show that this inequality holds true in full generality if one replaces C by CLK or Cdovr (K, BP nk ), where LK is the isotropic constant of K and dovr (K, BP nk ) is the outer volume ratio distance from K to the class of generalized k-intersection bodies. The main results are discussed in 3.4. Sections 6 and 7 outline briefly my dissertation work and my image processing projects, respectively. Section 4 contains a description of my current projects. There are many intriguing open problems in convex geometry and harmonic analysis that I hope to work on in the future. I will describe them in Section 5 and throughout the text. 2. Convex Geometry and Probability 2.1. Bounding marginal densities via affine isoperimetry. In [13] we establish connections between affine isoperimetric inequalities in convex geometry and concentration results for high dimensional probability distributions. Let µ be a probability measure on Rn with a bounded density f and let E be a k-dimensional subspace of Rn . The density of the marginal R πE (µ) on E is given by fπE (µ) (x) = E ⊥ +x f (y)dy, with x ∈ E. M. Rudelson and R. Vershynin Q [39] recently proved that if f (x) = ni=1 fi (xi ), where each fi is a bounded density on R, then for every 1 ≤ k ≤ n − 1 and every k-dimensional subspace E, (2.1)

kfπE (µ) k1/k ∞ ≤ C maxkfi k∞ i≤n

where C is a numeric constant and k·k∞ is the L∞ -norm. On the other hand, even for products, the stronger inequality (2.2)

1/n kfπE (µ) k1/k ∞ ≤ Ckf k∞

need not hold for all subspaces E. Indeed, set σ = (2π)−n/(2k) and E = span{e1 , . . . , ek }, let f be the Gaussian density with law µ = N (0, D), where D is the diagonal matrix D = n−k 1/k diag(σ 2 , . . . , σ 2 , 1, . . . , 1), with σ 2 repeated k times, then kf k∞ = 1, but kfπE (µ) k∞ = (2π) 2 . However, we are able to prove the following. The Haar probability measure on the Grassmann manifold Gn,k of all k-dimensional subspaces of Rn is denoted by µn,k . Theorem 2.1. Suppose µ is a probability measure on Rn with a bounded density f . Then for each 1 ≤ k ≤ n − 1 and s > 1, there exists As ⊆ Gn,k with µn,k (As ) ≥ 1 − 2s−kn such that: (i) for every E ∈ As and t > 1, there exists a set Bt ⊆ E such that πE (µ)(Bt ) ≤ t−kn and fπE (µ) (x)1/k ≤ c1 stkf k1/n ∞ ,

x ∈ (E \ Bt ) ∪ {0};

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(ii) for every E ∈ As , ε > 0 and any z ∈ E,  √  kn/(n+1) . πE (µ) {x ∈ E : |x − z| ≤ ε k} ≤ (c2 sεkf k1/n ∞ ) By means of the above example we show that the probability estimate for the µn,k -measure in Theorem 2.1 is sharp in each dimension k. One of the main ingredients are S. Szarek’s estimates for the entropy numbers on the Grassmannian [42]. Furthermore, one must exclude exceptional sets of positive πE (µ)-measure as can be seen by considering a neighborhood of a Besicovitch set. It can also be shown that for E ∈ / As there is E0 , close to E in the operator norm, such that E0 has a nearly optimal small-ball probability estimate. The second part of Theorem 2.1 follows from the first and the first part is established by an application of the Markov’s inequality to the following results. Below Mn,k denotes the affine Grassmannian. Theorem 2.2. Let 1 ≤ k ≤ n − 1 and f be a non-negative bounded integrable function on Rn . Then
n R k Z Z f (x)dx ωkn E (2.3) dE ≤ k f (x)dx . n−k kf |E k∞ ωn Gn,k Rn Z (2.4) Mn,k


n+1 k+1 Z f (x)dx ωkn+1 ωn(k+1) F . f (x)dx dF ≤ k+1 kf |F kn−k ωn ωk(n+1) Rn ∞

R

(2.3) holds for k different functions as well as for more general exponents. We also characterize the equality cases for both inequalities. Evaluating (2.3) for a characteristic function of a convex body K, f = 1K , we recover a result of H. Busemann - E. Straus [6] and E. Grinberg [18], the case k = n − 1 is Busemann’s seminal intersection inequality [5]. Evaluating (2.4) for a characteristic function of a convex body, gives back a result of R. Schneider [40]. R. Gardner [16] generalized the latter results, among other related inequalities, to the class of bounded, Borel measurable sets with a precise characterization of equality cases, making use of results due to R. Pfiefer [37], [38]. Our analysis of equality cases in (2.3) and (2.4) rests heavily on their results. Our interest in quantities such as the left-hand side of (2.3) stems from the following notion: for 1 ≤ k < n, the dual affine quermassintegrals of a convex body K ⊂ Rn are defined by !1/n Z ω n n e n−k (K) = Φ Voln (K ∩ E) dµn,k (E) . ωk Gn,k These were introduced by E. Lutwak and have proved to be an indispensable tool for quantitative questions concerning high-dimensional probability distributions [30, 35, 36]. In [18], e n−k (K) = Φ e n−k (gK) for each volume-preserving linear transformaE. Grinberg proved that Φ tion g. Motivated by Grinberg’s result, we prove that the quantities on the left-hand side of (2.3) and (2.4) are also invariant under volume preserving linear and affine transformations, respectively. Our argument uses the structure of semi-simple Lie groups.

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The main ingredients in the proof of Theorem 2.2 are the functional version of Busemann’s random simplex inequality from [34] and Blaschke-Petkantschin formulas. 3. Harmonic Analysis and Convex Geometry 3.1. The Busemann-Petty problem in the complex hyperbolic space. After the solution of the original Busemann-Petty problem, it was studied on other spaces as were its numerous generalizations. The interest in such studies stems from the fact that the geometric properties of intersection bodies are not known. I will mention just a few examples of such extensions that relate to my work. A. Zvavitch solved the Busemann-Petty problem on Rn for arbitrary measures, [45]. V. Yaskin studied the Busemann-Petty problem in real hyperbolic and spherical spaces, [43]. A. Koldobsky, H. K¨onig and M. Zymonopoulou demonstrated in [26] that the answer to the complex version of the Busemann-Petty problem is affirmative for the complex dimension n ≤ 3 and negative for n ≥ 4. Before we can formulate the Busemann-Petty problem in the complex hyperbolic space, we need some preparation. For ξ ∈ Cn with |ξ| = 1, denote by n

Hξ := {z ∈ C : (z, ξ) =

n X

zk ξk = 0}

k=1

the complex hyperplane through the origin perpendicular to ξ. We identify Cn with R2n via the mapping (3.1)

(ξ11 + iξ12 , . . . , ξn1 + iξn2 ) 7→ (ξ11 , ξ12 , . . . , ξn1 , ξn2 ) .

Under this mapping the hyperplane Hξ turns into a (2n − 2)-dimensional subspace of R2n . Recall that the Minkowski functional of a star body K is defined by kxkK := min{a ≥ 0 : x ∈ aK} , with x ∈ Rn . A convex body K in R2n is called Rθ -invariant, if for every θ ∈ [0, 2π] and every ξ = (ξ11 , ξ12 , . . . , ξn1 , ξn2 ) ∈ R2n (3.2)

kξkK = kRθ (ξ11 , ξ12 ), . . . , Rθ (ξn1 , ξn2 )kK ,

where Rθ stands for the counterclockwise rotation by an angle θ around the origin in R2 . We use the ball model of the n-dimensional complex hyperbolic space HnC . In this model n HC is identified with the interior of the open unit ball in Cn equipped with the Bergman metric. Bodies contained in the open unit ball that are geodesically convex with respect to the Bergman metric will be called h-convex. We say that origin-symmetric convex bodies in HnC are bodies that correspond under the mapping (3.1) to Rθ -invariant h-convex bodies in R2n . Denote the volume of a body K in R2n with respect to the Bergman metric by HVol2n (K). Now the Busemann-Petty problem in HnC can be formulated as follows. Given two Rθ invariant h-convex bodies K and L in R2n such that HVol2n−2 (K ∩ Hξ ) ≤ HVol2n−2 (L ∩ Hξ )

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for any ξ ∈ S2n−1 , does it follow that HVol2n (K) ≤ HVol2n (L)? In [9], we provide the positive answer to this problem for n = 2 and show that for n ≥ 3 the answer is negative. The main steps in the proof are the following: - Use the counterexamples to the Busemann-Petty problem in Cn with n ≥ 4 from [26] to construct counterexamples to the Busemann-Petty problem in HnC for n ≥ 4. - Express the volume of sections with respect to the Bergman metric in Fourier analytic terms. - Establish a connection between the Busemann-Petty problem in HnC and the distribution kxk−2 K

2

1−(|x|kxk−1 K )

, analogous to the Lutwak’s connection described in the introduction. Note that

an origin symmetric star body K in Rn is an intersection body if and only if k · k−1 K is a n positive definite distribution on R , see Theorem 4.1 in [25]. - Provide an affirmative answer to the problem for n = 2, by proving that for any Rθ -invariant star body contained in the open unit ball the distribution of interest is positive definite. - Provide a negative answer to the problem for n = 3, by constructing a counterexample of an Rθ -invariant h-convex body contained in the open unit ball whose distribution of interest is not positive definite. This finishes the proof. Our proof uses methods from [26] and [43], as well as results for complex star bodies from [27]. 3.2. The lower-dimensional Busemann-Petty problem in the complex hyperbolic space. It is natural to ask what happens if hyperplane sections are replaced by sections of lower dimensions in the Busemann-Petty problem. Let 1 ≤ k ≤ n − 2 and let K and L be Rθ -invariant h-convex bodies in R2n . Suppose that for every complex linear subspace H of complex dimension k HVol2k (K ∩ H) ≤ HVol2k (L ∩ H) . Does it follow that HVol2n (K) ≤ HVol2n (L)? In [10], we show that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions. The proof uses previously obtained results about k-intersection bodies. This class of bodies in Rn was introduced by A. Koldobsky in [23, 24]. He proved that an origin-symmetric star body K in Rn is a k-intersection body if and only if k · k−k K represents a positive definite distribution. There are examples of Rθ -invariant convex bodies in R2n , n ≥ 3, that are not k-intersection bodies for any 1 ≤ k ≤ 2n−4. Namely, the unit balls Bqn , q > 2, of the complex space lqn considered as subsets of R2n : q

q

Bqn = {x ∈ R2n : (x211 + x212 ) 2 + · · · + (x2n1 + x2n2 ) 2 ≤ 1} , see Theorem 4 in [26]. The two main cases to consider are: k = 2 and 3 ≤ k ≤ n − 1. Since one-dimensional sections are discs, the affirmative answer in this case is immediate. For the case 3 ≤ k ≤ n − 1

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we construct counterexamples to the problem starting with Rθ -invariant convex bodies in R2n that are not 2(n−k)-intersection bodies. For the case k = 2 we first construct an Rθ -invariant h-convex body ‘by hand’ and then use it to construct a counterexample to the problem. Note that the lower-dimensional Busemann-Petty problem is still open on Rn with n ≥ 5 for sections by subspaces of dimension 2 and 3. This problem in Rn is related to a generalization of the class of intersection bodies introduced by G. Zhang in [44] as follows. For 1 ≤ k ≤ n−1, let G(n, n − k) be the Grassmanian of (n − k)-dimensional subspaces of Rn . The (n − k)dimensional spherical Radon transform is an operator Rn−k : C(Sn−1 ) → C(G(n, n − k)) defined by Z Rn−k f (H) = f (x)dx , Sn−1 ∩H

for H ∈ G(n, n − k). Denote the image of the operator Rn−k by X and the space of positive linear functionals on X by M + (X). An origin-symmetric star body K in Rn is called a generalized k-intersection body, K ∈ BP nk , if there exists a functional ν ∈ M + (X) so that for every f ∈ C(Sn−1 ) Z Sn−1

kxk−k K f (x)dx = ν(Rn−k f ) .

The connection between the two generalizations of intersection bodies is the following: Every generalized k-intersection body is a k-intersection body, see [24], and this inclusion is strict, see [31]. 3.3. Sections of convex bodies with symmetries. Intersection bodies were introduced by E. Lutwak in 1988 in his celebrated paper [28] in connection with the Busemann-Petty problem. We recall that an origin-symmetric star body K in Rn is the intersection body of an origin-symmetric star body L if the radius of K in every direction equals to the (n − 1)dimensional volume of the central hyperplane section of L perpendicular to this direction. In other words, for every unit vector ξ in Rn , (3.3)

⊥ kξk−1 K = Voln−1 (L ∩ ξ ) ,

where ξ ⊥ is the hyperplane perpendicular to ξ. Using polar coordinates, equation (3.3) becomes Z 1 1 −1 kξkK = kθk−n+1 dθ = )(ξ) . Rn−1 (k · k−n+1 L L n − 1 Sn ∩ξ⊥ n−1 Hence, a star body K in Rn is the intersection body of a star body if and only if k · k−1 K is the spherical Radon transform of a continuous positive function on Sn−1 . A more general class of intersection bodies in Rn was introduced by P. Goodey, E. Lutwak and W. Weil in 1996 in [17]. A star body K is an intersection body if there exists a finite non-negative Borel measure µ on the sphere so that k · k−1 K = Rn−1 µ. An analogous class of bodies in Cn was studied by A. Koldobsky, G. Paouris and M. Zymonopoulou in [27]. A complex convex body K in Cn is a convex body in R2n that is invariant under the block diagonal subgroup of SO(2n) of the form {diag(g, . . . , g) : g ∈ SO(2)} ,

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where SO(·) stands for the special orthogonal group over the reals, compare to (3.2). Intersection bodies in Cn were defined along the same lines as intersection bodies in Rn , taking into account the above invariance. They inherit many properties of their real counterparts. In [14], we studied intersection bodies in Rκn that are invariant under the block diagonal subgroup of SO(κn) of the form {diag(g, . . . , g) : g ∈ SO(κ)} , where κ ∈ N is fixed. Call subsets of Rκn that satisfy the above invariance κ-balanced and denote by Kn the space Rκn with the property that all geometric objects in this space (such as star shaped bodies, linear subspaces, etc.) satisfy the above invariance. For κ = 1, 2, 4, Kn can be thought of as the n-dimensional real, complex or quaternionic vector space, respectively; however our results hold in more generality for any κ ∈ N. Following ideas from [27], we generalize to Kn many known results from the theory of intersection bodies in Rn and Cn . Our main results are: - Intersection bodies in Kn coincide with two generalizations of the class of real intersection bodies due to A. Koldobsky and G. Zhang: the κ-balanced κ-intersection bodies in Rκn and κ-balanced generalized κ-intersection bodies in Rκn . - The above fact allows to extend to Kn the result of P. Goodey and W. Weil that intersection bodies in Rn can be obtained as the closure in the radial metric of radial sums of ellipsoids. - The Busemann-Petty problem in Kn for arbitrary measures. From the stability consideration in this problem a series of interesting inequalities is derived. - Extension of the Busemann’s theorem to intersection bodies of convex bodies in Kn : The intersection body of a convex body in Kn is convex. 3.4. On the average volume of sections of convex bodies. The average section functional as(K) of a centered set in Rn is the average volume of the central hyperplane sections of K: Z Voln−1 (K ∩ ξ ⊥ ) dσ(ξ).

as(K) = S n−1

A set is called centered if its barycenter lies at the origin. It was proved in [21] that if K is an intersection body in Rn , then 1 ωn−1 ⊥ as(K) 6 bn,1 Voln (K) n max as(K ∩ ξ ), with b ' 1. n,1 := 1/n ξ∈S n−1 ωn−2 ωn Whenever we write a . b we mean that there exists an absolute constant c > 0 such that a 6 cb, and whenever we write a ' b, we mean that a . b and b . a. In this project we discuss similar inequalities for as(K) for an arbitrary centered convex body K in Rn . More precisely, we study the following question. Question 3.1. Let 1 6 k < n and define γn,k as the smallest constant γ > 0 for which the following holds true: for every centered convex body K in Rn we have (3.4)

k

as(K) 6 γ k Voln (K) n

max

E∈Grn,n−k

as(K ∩ E).

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Is it true that supn,k γn,k < ∞? First, we prove that (3.4) holds true for the class of origin-symmetric star bodies in Rn if one replaces γ by γdovr (K, BP nk ), where ( ) 1/n Vol (D) n dovr (K, BP nk ) = inf : K ⊂ D, D ∈ BP nk Voln (K) stands for the outer volume ratio distance from K to the class BP nk of generalized k-intersection bodies. More precisely, in this case ωn−1 ' 1. γn,k 6 bn,k dovr (K, BP nk ), where bkn,k = k/n ωn−k−1 ωn For K origin-symmetric and convex, the distance dovr (K, BP nk ) was estimated in [22] implying that γn,k is bounded by a function of n/k, and hence it remains bounded as long as k is proportional to n. For certain classes of origin-symmetric convex bodies the distance dovr (K, BP nk ) is bounded by an absolute constant. Restricting to any of these classes provides an affirmative answer to Question 3.1. Second, we prove that (3.4) holds true for centered convex bodies if one replaces γ by γLK , where LK is the isotropic constant of K. A centered convex body K of volume 1 in Rn is called isotropic if there exists a constant LK > 0 such that Z hx, ξi2 dx = L2K K n−1

for every ξ ∈ S . Every centered convex body K has an isotropic position T (K), T ∈ GL(n), which is uniquely defined modulo orthogonal transformations. A well-known question in asymptotic convex geometry asks if there exists an absolute constant C > 0 such that LK 6 C for every n and every centered convex body K in Rn . The best known upper bound √ Ln := sup{LK : K isotropic in Rn } 6 c 4 n is due to Klartag [20]. It is known that for many classes of convex bodies the isotropic constant LK is bounded by an absolute constant, restricting to these classes provides an affirmative answer to Question 3.1. We also show that γn,k . γn,1 ' Ln for all 1 6 k 6 n − 1. When k is proportional to n the situation is analogous to the first case discussed above. Moreover, for the mean value of the average section functional as(K ∩ E) over all E ∈ Grn,n−k we obtain the next general upper and lower bounds: Let K be a centered convex body in Rn 1 and define p(K) := R(K)/Voln (K) n , where R(K) is the circumradius of K. Then, for every 1 6 k 6 n − 1 we have that  0  k  √ k Z k c n c p(K) n−1 √ (3.5) as(K) 6 Voln (K) n as(K ∩ E) dνn−k (E) 6 as(K), p(K) n Grn,n−k where c, c0 > 0 are absolute constants.

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The methods that are used for the two cases are independent. The first method allows us to work with origin-symmetric (not necessarily convex) star bodies while the second method allows us to work with (not necessarily symmetric) centered convex bodies. 3.5. Busemann’s intersection inequality in hyperbolic and spherical spaces. Busemann’s intersection inequality (1.1) can also be stated as follows: Centered ellipsoids in Rn are the only maximizers of the quantity Z (3.6) Voln−1 (K ∩ θ⊥ )n dσ(θ) Sn−1

in the class of star bodies of a fixed volume. In [11] we study this question in the real hyperbolic space Hn and the real spherical space Sn+ . We show that in Hn centered balls are the unique maximizers of (3.6) in the class of star bodies of a fixed volume. On the sphere the situation is different. In S2+ centered balls are in fact the unique minimizers (in the class of origin-symmetric star bodies). The maximizers of (3.6) in the class of origin-symmetric star bodies in S2+ are cones. The maximizers of (3.6) in the class of origin-symmetric convex bodies in S2+ are lunes. It is surprising that in Sn+ with n ≥ 3 centered balls are neither maximizers nor minimizers, even in the class of originsymmetric convex bodies. Moreover, we obtain an optimal lower bound for (3.6) in the class of star bodies in Sn+ , n ≥ 3, of a given volume and describe the equality cases. We prove a version of Busemann’s intersection inequality (together with the equality cases) for general n n measures on Rn and Hn . More precisely, let µ be a measure with a radially R on R or H symmetric and decreasing density. Then the maximum of S n−1 µ(K ∩ ξ ⊥ )n dξ in the class of star bodies of a fixed measure µ is given by the geodesic balls centered at the origin. An important special case is that of the Gaussian measure on Rn . It is interesting to note that, in the context of the Busemann-Petty problem, the sphere and the Euclidean space are similar in the sense that the positive answer holds in the same dimensions, while the hyperbolic space exhibits a different behaviour; see [43]. For Busemann’s intersection inequality, the hyperbolic space is similar to the Euclidean space, while the sphere is not. It is also worth mentioning that the answer to the Busemann-Petty problem for arbitrary measures on Rn is the same as in the case of volume; see [45]. Likewise, Busemann’s intersection inequality for general measures on Rn or Hn is similar to that for volume. 4. Current Work 4.1. Maximum area polygons circumscribed about a convex polygon. This project is a collaboration with Markus Aussenhofer from the University of Vienna, Zsolt L´angi from the Budapest University of Technology and G´eza T´oth from the Hungarian Academy of Sciences. A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We address the following question: given a convex polygon P , find a maximal area convex polygon circumscribed about P . We investigate the properties of these

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polygons, and present an algorithm to construct them. For the case of regular n-gons, we have an explicit solution. In particular, we disprove Farris’s conjecture [15]. This question is related to finding the best upper bound estimate of the Gini index, originally used in statistics to measure the inequality in income amongst households. To every good and society one can define a distribution curve L : [0, 1] → [0, 1] such that L(p) is the fraction of the total good that is controlled by the poorest fraction p of the society. Functions with this property are commonly referred to as Lorenz Curves. We call any convex and non-negative curve L : [0, 1] → [0, 1] with L(0) = 0 and L(1) = 1 a Lorenz curve. Note that the curve L(p) = p corresponds to the equal distribution: the poorest p percent earn p percent of the the total good, i.e. everyone controls the same amount. The Gini index G(L) is defined as the ratio of the area between the identity and the Lorenz curve to the area below the identity. In practice, only some points of the Lorenz curve can be obtained through f.e. surveys and one is interested to find the upper and lower bounds for the Gini index from the available data. The best lower bound follows easily from the convexity of a Lorenz curve: it is attained at the polygonal curve connecting the known points of the curve. Our work provides an algorithmic solution to the question of finding the best upper bound corresponding to any given point set. 4.2. Smooth valuations. This project is a collaboration with Judit Abardia and Andreas Bernig from Goethe University Frankfurt. Let V be an Euclidean vector space of dimension n and let K(V ) be the space of non-empty compact convex subsets in V . A valuation on V is a map µ : K(V ) → R satisfying µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L), whenever K, L, K ∪ L ∈ K(V ). We say that µ is continuous if it is so with respect to the topology on K(V ) induced by the Hausdorff metric. A flag area measure on V is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold Flag1,p+1 consisting of a unit vector v and a (p + 1)-dimensional linear subspace containing v, where 0 ≤ p ≤ n − 1. Using local parallel sets Hinderer constructed examples of SO(n)-invariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes involving the squared cosine of the angle between two subspaces. We construct a more general space of SO(n)-invariant flag area measures via integration of appropriate differential forms. We compute the dimension of this space, discuss their properties and provide explicit formulas on polytopes, which are similar to the formulas for Hinderer’s examples, however with an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Hinderer’s flag area measures correspond to special cases where the elementary symmetric polynomial is just the product. Moreover, we construct an explicit basis for this space, which gives a classification result in the spirit of Hadwiger’s theorem. 4.3. Isoperimetric inequalities. This is a continuing collaboration with Grigoris Paouris from the Texas A&M University and Peter Pivovarov from the University of Missouri.

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We work on the extensions of our results from [13] to flag manifolds. Another related project stems from the invariance results obtained in [13]. The linear/affine invariance holds for more general geometric objects than those in Theorem 2.2 and suggests the existence of extremizers and new inequalities for these objects. 5. Future Research Plans Problem 5.1. In the light of Theorem 2.1, a natural question arises: under what additional conditions, can one guarantee that all marginal densities of such functions f are suitably bounded, i.e. 1/n kfπE (µ) k1/k ∀E ∈ Gn,k . ∞ ≤ Ckf k∞ , R R One may assume that µ is isotropic, i.e. Rn hx, θidµ(x) = 0 and Rn |hx, θi|2 dµ(x) = 1 for θ ∈ S n−1 . If µ is an isotropic, subgaussian, log-concave probability measure, then (5.1) holds as a consequence of a result of J. Bourgain on the isotropic constant of such measures [3]. 1/n If µ is isotropic and log-concave, the isotropic constant of µ is defined by Lµ := kfµ k∞ . A major open problem, known as the Hyperplane Conjecture, asks if there exists an absolute constant C such that Lµ ≤ C. The best known bound (of order n1/4 ) is due to B. Klartag [20], improving an earlier result of J. Bourgain [2]. Thus in the class of isotropic log-concave probability measures µ, inequality (5.1) amounts to asking if

(5.1)

LπE (µ) ≤ CLµ , ∀E ∈ Gn,k .

(5.2)

The above question is another equivalent formulation of the Hyperplane Conjecture, see [33]. Problem 5.2. One can interpret (2.4) as an inequality about the k-plane transform. Recall that the k-plane transform Tn,k applied to a function f on Rn is defined by Z Tn,k (f )(F ) = f (x)dx (F ∈ Mn,k ). F

When k = n − 1, Tn,k is the Radon transform and when k = 1, it is the X-ray transform. The k-plane transform satisfies several key inequalities. In particular, for each q ∈ [1, n + 1], there is a unique p ∈ [1, (n + 1)/(k + 1)] such that kTn,k (f )kq ≤ C(n, k, q)kf kp p

for all f ∈ L , [7]. See [1] for related work and a conjecture about the extremal functions. Is there a connection between our results and results about the k-plane transform? Problem 5.3. The answer to the original Busemann-Petty problem is affirmative for intersection bodies. Can one find an analog for the real/complex intersection body in the hyperbolic setting? Problem 5.4. The lower-dimensional Busemann-Petty problem in Cn . The proof in [10] can be adjusted to show the negative answer for the case of sections of complex dimension 3 ≤ k ≤ n − 2. The only open case is k = 2. Moreover, what are the analogs of complex generalized k-intersection bodies?

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Problem 5.5. Shephard’s problem in Rn for arbitrary measures. Complex version of this problem. Problem 5.6. Slide the center of a small ball around a closed curve, measure the length of the two curve segments inside the ball. Suppose that the two length are always equal to each other, is this curve the boundary of a ball? 6. Paley-Wiener type theorems on symmetric spaces Real life signals are functions. They are usually defined on certain sets or manifolds. What one measures is often not the function itself, but some transform of the function. For example frequency properties of a signal correspond to the Fourier transform of that signal. The Xray tomography is expressed as averages over lines, or more generally, as averages over lower dimensional planes. This leads to the Radon transform and its inversion. The questions are always the same: can we reconstruct the original signal, or can we at least derive enough properties of the original signal, from the available measurements. The first problem equates to finding an inversion formula for the given transform. The second problem corresponds to sampling theorems or to characterization of function spaces in terms of their image under the transform in question. One fundamental property of the human body and other natural objects is that they are finite. This is expressed in the assumption that the signals are compactly supported. The Paley-Wiener type theorems describe the image of compactly supported functions or distributions under an integral transformation. The intimate relationship between the support properties of a function on Rn and the analyticity of its Fourier transform was first analyzed by R. Paley and N. Wiener in [32], where they considered square-integrable functions. It was extended by L. Schwartz [41] to distributions and by L. H¨ormander [19] to smooth compactly supported functions. Signals with compactly supported Fourier transform are called bandlimited. By the NyquistShannon sampling theorem bandlimited signals can be reconstructed perfectly from their samples, given that the sampling rate is more than twice the maximum frequency. Therefore Paley-Wiener type theorems are related to the sampling theory, as they describe bandlimited signals. The Fourier transform on Rn interchanges spatial information and frequency information. It is naturally related to the translation invariance of measurements and convolution. Continuous symmetries of a physical system are described by Lie groups. Given a sufficiently good Lie group G, there is a natural Fourier transform on L2 (G) coming from the representations of G. The Fourier transform in this more general setting is operator or vector valued. The Euclidean motion group SO(n) n Rn is the group of symmetries of a classical nonrelativistic ´ physical system. Together with G. Olafsson we described the image of smooth compactly supported scalar valued and Hilbert space valued functions under the Fourier transform that is naturally associated to the Euclidean motion group [12]. I will list our main results below. We begin by extending the classical Paley-Wiener theorem to functions with values in a separable Hilbert space (under the usual Fourier transform on Rn ); and examine the special

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case of the Hilbert space of square integrable functions on the sphere that are SO(n)-finite. Then we work with a more general Fourier transform on Rn . Let G denote the Euclidean motion group. (G, SO(n)) is a Gelfand pair. In particular, Rn ' G/SO(n). This realization of Rn comes with its own natural Fourier transform derived from the representation theory of G, call it FG . The representations of G that contribute to the decomposition of L2 (Rn ) are parameterized by R+ . We describe the spaces of smooth compactly supported functions and Schwartz functions on Rn under FG . Finally, we discuss extensions of the classical PaleyWiener theorem as well as of our description of the space of smooth compactly supported functions on Rn to projective limits of the corresponding function spaces. This can also be interpreted as an extension to the inductive limits of the underlying spaces. My dissertation is accessible through the LSU electronic dissertation library. 7. Image processing The advances in the camera technologies and a rapid increase in computational power opened the door for image processing techniques in the area of automated optical quality control. For a number of reasons this alternative is very attractive for quality control of welded seams. The laser weld seams have a characteristic Chevron pattern possessing rich spectral information, suggesting the use of multiresolution analysis for its analysis and segmentation. In my undergraduate thesis project Texture analysis with Gabor filters I explored the applicability of Gabor filters for texture analysis of laser weld seams. My work was guided by U. M¨ ussigman in partnership with the Swiss company Soudronic. My thesis is readily available on my website. It is written in German. ˇ c in partnership with S. Bujenovi´c from the Lake P.E.T. Imaging Together with S. Zabi´ Center, we worked on the improved readability of PET/CT images. To see the details in lungs, bones or soft tissue, CT images have to be scaled to the according range of gray values. Thus, when evaluating CT images physicians have to look through the same set of images more than twice. We worked on a CT image that shows details in all tissues simultaneously overlaid with the relevant PET information. In the framework of this project I also led undergraduate research projects in image processing. References 1. Albert Baernstein, II and Michael Loss, Some conjectures about Lp norms of k-plane transforms, Rend. Sem. Mat. Fis. Milano 67 (1997), 9–26 (2000). 2. Jean Bourgain, On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. 3. , On the isotropy-constant problem for “PSI-2”-bodies, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 114–121. 4. Silouanos Brazitikos, Susanna Dann, Apostolos Giannopoulos, and Alexander Koldobsky, On the average volume of sections of convex bodies, to appear in the Israel Journal of Mathematics. 5. Herbert Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1–12. 6. Herbert Busemann and Ernst G. Straus, Area and normality, Pacific J. Math. 10 (1960), 35–72. 7. Michael Christ, Estimates for the k-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910. 8. Susanna Dann, On the Minkowski-Funk transform, arXiv:1003.5565.

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9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

33.

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, The Busemann-Petty Problem in the Complex Hyperbolic Space, Mathematical Proceedings of the Cambridge Philosophical Society 155 (2013), 155–172. , The lower dimensional Busemann-Petty problem in the complex hyperbolic space, Adv. in Appl. Math. 53 (2014), 44–60. Susanna Dann, Jaegil Kim, and Vladyslav Yaskin, Busemann’s intersection inequality in hyperbolic and spherical spaces, to appear in Advances in Mathematics. ´ Susanna Dann and Gestur Olafsson, Paley-Wiener theorems with respect to the spectral parameter, New developments in Lie theory and its applications, Contemp. Math., vol. 544, Amer. Math. Soc., Providence, RI, 2011, pp. 55–83. Susanna Dann, Grigoris Paouris, and Peter Pivovarov, Bounding marginal densities via affine isoperimetry, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 140–162. Susanna Dann and Marisa Zymonopoulou, Sections of convex bodies with symmetries, Adv. Math. 271 (2015), 112–152. Frank A. Farris, The Gini index and measures of inequality, Amer. Math. Monthly 117 (2010), no. 10, 851–864. Richard J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities, Adv. Math. 216 (2007), no. 1, 358–386. Paul Goodey, Erwin Lutwak, and Wolfgang Weil, Functional analytic characterizations of classes of convex bodies, Math. Z. 222 (1996), no. 3, 363–381. Eric L. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies, Math. Ann. 291 (1991), no. 1, 75–86. Lars H¨ ormander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press Inc., New York, 1963. Boaz Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1274–1290. A. Koldobsky, Stability and separation in volume comparison problems, Math. Model. Nat. Phenom. 8 (2013), no. 1, 156–169. A. Koldobsky, G. Paouris, and M. Zymonopoulou, Isomorphic properties of intersection bodies, J. Funct. Anal. 261 (2011), no. 9, 2697–2716. Alexander Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), no. 4, 827–840. , A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000), no. 6, 1507–1526. , Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. Alexander Koldobsky, Hermann K¨ onig, and Marisa Zymonopoulou, The complex Busemann-Petty problem on sections of convex bodies, Adv. Math. 218 (2008), no. 2, 352–367. Alexander Koldobsky, Grigoris Paouris, and Marisa Zymonopoulou, Complex intersection bodies, J. Lond. Math. Soc. (2) 88 (2013), no. 2, 538–562. Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. , Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, NorthHolland, Amsterdam, 1993, pp. 151–176. Emanuel Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), no. 2, 566–598. , Generalized intersection bodies are not equivalent, Adv. Math. 217 (2008), no. 6, 2822–2840. Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1934. Grigoris Paouris, On the isotropic constant of marginals, Studia Math. 212 (2012), no. 3, 219–236.

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34. Grigoris Paouris and Peter Pivovarov, A probabilistic take on isoperimetric-type inequalities, Adv. Math. 230 (2012), no. 3, 1402–1422. 35. , Small-ball probabilities for the volume of random convex sets, Discrete Comput. Geom. 49 (2013), no. 3, 601–646. 36. Grigoris Paouris and Petros Valettas, Neighborhoods on the Grassmannian of marginals with bounded isotropic constant, J. Funct. Anal. 267 (2014), no. 9, 3427–3443. 37. Richard E. Pfiefer, The extrema of geometric mean values, ProQuest LLC, Ann Arbor, MI, 1982, Thesis (Ph.D.)–University of California, Davis. , Maximum and minimum sets for some geometric mean values, J. Theoret. Probab. 3 (1990), 38. no. 2, 169–179. 39. Mark Rudelson and Roman Vershynin, Small ball probabilities for linear images of high-dimensional distributions, Int. Math. Res. Not. IMRN (2015), no. 19, 9594–9617. 40. Rolf Schneider, Inequalities for random flats meeting a convex body, J. Appl. Probab. 22 (1985), no. 3, 710–716. 41. Laurent Schwartz, Transformation de Laplace des distributions, Comm. S´em. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] (1952), no. Tome Supplementaire, 196–206. 42. Stanislaw J. Szarek, The finite-dimensional basis problem with an appendix on nets of Grassmann manifolds, Acta Math. 151 (1983), no. 3-4, 153–179. 43. Vladyslav Yaskin, The Busemann-Petty problem in hyperbolic and spherical spaces, Adv. Math. 203 (2006), no. 2, 537–553. 44. Gaoyong Zhang, Sections of convex bodies, Amer. J. Math. 118 (1996), no. 2, 319–340. 45. Artem Zvavitch, The Busemann-Petty problem for arbitrary measures, Math. Ann. 331 (2005), no. 4, 867–887.