相关论文: An isomorphic version of the slicing problem

For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…

Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding…

The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by…

We provide an affirmative answer to a problem posed by Barvinok and Veomett, showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of a sub-exponential dimension. Moreover, we…

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary…

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally…

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le…

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${\mathbb…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…

Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R^n. We prove that, for every $\delta>0$, the isotropy constant of K is bounded by a constant $c(\delta)$ with high probability,…

In this paper, we begin by constructing global linear maps on (n-2)-dimensional subspaces, derived from the local continuity of linear transformations among central sections of a convex body. Using these linear maps, we subsequently…

It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb R^n$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$ of…

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{\xi\in S^{n-1}} \int_{K\cap \xi^\bot} f $$ for all centrally-symmetric convex bodies $K$ in…

In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated…

We determine when a convex body in $\mathbb{R}^d$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l},$ $d=d_1\cdots d_l.$ We call these convex bodies "tensorial bodies". We prove that,…

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $\rho$. We show…

Let K be a closed bounded convex subset of $\Bbb R^n$; then by a result of the first author, which extends a classical theorem of Whitney there is a constant $w_m(K)$ so that for every continuous function f on K there is a polynomial $\phi$…