arXiv subjects
Publications and source records attributed to Susanna Dann.
We prove that a convex polytope $P \subset \mathbb{R}^d$, $d \ge 2$, of uniform density $\delta \in (0,1)$ floating in a liquid of density $1$, is uniquely determined by its surface of flotation $P_{[\delta]}$ whenever $\delta \neq \tfrac{1}{2}$. Analogously, we show that the buoyancy surface $\mathcal{C}_\delta P$ of a convex polytope $P$ with prescribed density $\delta \in (0,1)$ uniquely determines $P$.
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 present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time. As an application, we disprove a conjecture of Farris. Moreover, for the special case of regular n-gons we find an explicit solution.
A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0 \leq p \leq n-1$. Using local parallel sets, Hinderer constructed examples of $\mathrm{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\mathrm{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\mathrm{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger's theorem.
Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.
Busemann's intersection inequality asserts that the only maximizers of the integral $\int_{S^{n-1}} |K\capξ^\perp|^n dξ$ among all convex bodies of a fixed volume in $\mathbb R^n$ are centered ellipsoids. We study this question in the hyperbolic and spherical spaces, as well as general measure spaces.
The average section functional ${\rm as}(K)$ of a centered convex body in ${\mathbb R}^n$ is the average volume of central hyperplane sections of $K$: \begin{equation*}{\rm as}(K)=\int_{S^{n-1}}|K\cap ξ^{\perp }|\,dσ(ξ).\end{equation*} We study the question if there exists an absolute constant $C>0$ such that for every $n$, for every centered convex body $K$ in ${\mathbb R}^n$ and for every 0<k<n, $${\rm as}(K)\ls C^k|K|^{\frac{k}{n}}\,\max_{E\in {\rm Gr}_{n-k}}{\rm as}(K\cap E).$$ We observe that the case $k=1$ is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces $C$ by $CL_K$ or $Cd_{\rm ovr}(K,{\cal{BP}}_k^n)$, where $L_K$ is the isotropic constant of $K$ and $d_{\rm ovr}(K,{\cal{BP}}_k^n)$ is the outer volume ratio distance from $K$ to the class ${\cal{BP}}_k^n$ of generalized $k$-intersection bodies. We also compare ${\rm as}(K)$ to the average of ${\rm as}(K\cap E)$ over all $k$-codimensional sections of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions as well as the natural lower dimensional analogue of the average section functional.
Let $μ$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann--Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $μ$, any marginal $π_E(μ)$, or a small perturbation thereof, satisfies a nearly optimal small-ball probability.
In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing complex and quaternionic convex bodies. We conduct a systematic study of sections of bodies with symmetries of this type, with the emphasis on problems of the Busemann-Petty type and hyperplane inequalities. The main role belongs to the class of intersection bodies with symmetries.
The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for k=2,3. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.
The Busemann-Petty problem asks whether origin-symmetric convex bodies in real Euclidean n-space with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative for n less or equal to 4 and negative if n greater or equal to 5. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative for n less or equal to 2 and negative for n greater or equal to 3.
One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of the set of irreducible representations in the support of the Plancherel measure for L^2(M). We then discuss the Euclidean motion group, semisimple symmetric spaces, and some limits of those spaces.
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
The subject of this paper is the history of the Minkowski-Funk Transform. After introducing the Minkowski-Funk Transform as well as its dual transform and a generalization of both, we will present an inversion formula of the Minkowski-Funk Transform. Then we will discuss the history of this problem: related work by Minkowski and Funk and the connection between their work.