Related papers: On the shape of a convex body with respect to its …

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…

Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper…

An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new…

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that…

We show that for any compact convex set $K$ in $\mathbb{R}^d$ and any finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy of $K$…

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the…

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…

For every convex body $K \subset \mathbb R^n$ and $\delta \in (0,1)$, the $\delta$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq \delta \left|K\right|_n$. We show that for $n=2$ and…

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…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…

The simplex was conjectured to be the extremal convex body for the two following "problems of asymmetry":\\ P1) What is the minimal possible value of the quantity $\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies…

The section volume function $A_K(\xi,t), \ \xi \in \mathbb R^n, \ t \in \mathbb R,$ of a body $K \subset \mathbb R^n$ evaluates the $(n-1)$-dimensional volume of the cross-section $K$ by the hyperplane $\{ x \cdot \xi=t \}.$ We are…

We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp…

We investigate Minkowski additive, continuous, and translation invariant operators $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $\Phi(K)$ is bounded from above and below by…

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a…

We construct examples of two convex bodies $K,L$ in $\mathbb{R}^n$, such that every projection of $K$ onto a $(n-1)$-dimensional subspace can be rotated to be contained in the corresponding projection of $L$, but $K$ itself cannot be…

For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the…

The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of…

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…