相关论文: On unbalanced difference bodies and Godbersen's co…

Let $K$ be a convex compact $GB$-subset of a separable Hilbert space $H$. Denote by $\mathrm{Spec}_k K$ the set $\{(\xi_1(h), \ldots, \xi_k(h))\colon h\in K\}\subset \mathbb{R}^k,$ where $\xi_1, \ldots, \xi_k$ are independent copies of the…

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between…

We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on…

This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the…

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $\Omega$. The bound depends only on the perimeter and inradius $r$ of the original body and states that \[|\partial\Omega_t| \geq…

A comparison problem for volumes of convex bodies asks whether inequalities $f_K(\xi)\le f_L(\xi)$ for all $\xi\in S^{n-1}$ imply that $\vol_n(K)\le \vol_n(L),$ where $K,L$ are convex bodies in $\R^n,$ and $f_K$ is a certain geometric…

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…

We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-\beta}$, for every…

Recall that a convex body $K$ is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in $K$. Approximating convex body in John's position by polytopes we obtain the following results. 1. Let $n>R_n\ge 1$…

Let $K$ and $K_0$ be convex bodies in $\mathbb{R}^d$, such that $K$ contains the origin, and define the process $(K_n, p_n)$, $n \geq 0$, as follows: let $p_{n+1}$ be a uniform random point in $K_n$, and set $K_{n+1} = K_n \cap (p_{n+1} +…

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an…

An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a signed…

In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex body $A\subset\mathbb{R}^{n}$ and its…

Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…

Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M…

We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding…

The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \geq 0$ such that $A$ is contained in a translate of $\lambda K$. Each choice of $K$ gives a…

It is proved that if $C$ is a convex body in ${\Bbb R}^n$ then $C$ has an affine image $\widetilde C$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection, $$|P\widetilde C| \ge |\widetilde C|^{n-1\over n}.$$ It…

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of…

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation…