Related papers: Convex bodies with few faces

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a…

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings,…

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u…

In a $d$-dimensional convex body $K$, for $n \leq d+1$, random points $X_0, \dots, X_{n-1}$ are chosen according to the uniform distribution in $K$. Their convex hull is a random $(n-1)$-simplex with probability $1$. We denote its…

We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme…

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the…

We obtain upper and lower bounds on the difference between the renormalized volume and the volume of the convex core of a convex cocompact hyperbolic 3-manifold which depend on the injectivity radius of the boundary of the universal cover…

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…

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, $k$-homogeneous valuation on convex bodies in $\mathbb{R}^n$ can be approximated…

We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both…

Let $K$ be an $n$-dimensional convex body. Define the difference body by $$ K-K= \{x-y \mid x,y \in K \}. $$ We estimate the volume of the section of $K-K$ by a linear subspace $F$ via the maximal volume of sections of $K$ parallel to $F$.…

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We…

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of…

Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the…

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…

In 2021, Ordentlich, Regev and Weiss made a breakthrough that the lattice covering density of any $n$-dimensional convex body is upper bounded by $cn^{2}$, improving on the best previous bound established by Rogers in 1959. However, for the…

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 prove that for any $n\in \mathbb{N}$ there is a convex body $K\subseteq \mathbb{R}^n$ whose surface area is at most $n^{\frac12+o(1)}$, yet the translates of $K$ by the integer lattice $\mathbb{Z}^n$ tile $\mathbb{R}^n$.