Related papers: Comments on the floating body and the hyperplane c…

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P.…

The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the…

In 1951, Bang posed the affine plank conjecture, which remains open: If a convex body in $\mathbb{R}^d$ is covered by planks, then the total relative width of the planks is at least one. We prove a lower bound of $2/(1+\sqrt{d})$ for this…

For a pseudoconvex tube domain, we prove estimates that relate the sublevel sets of its diagonal Bergman kernel to the floating bodies of its convex base. This allows us to associate a new affine invariant to any convex body.

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…

We compare and combine two approaches that have been recently introduced by Dafnis and Paouris [DP] and by Klartag and Milman [KM] with the aim of providing bounds for the isotropic constants of convex bodies. By defining a new hereditary…

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…

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…

The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the…

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…

This paper addresses the floating body problem which consists in studying the interaction of surface water waves with a floating body. We propose a new formulation of the water waves problem that can easily be generalized in order to take…

The second and fourth authors have conjectured that a certain hollow tetrahedron $\Delta$ of width $2+\sqrt2$ attains the maximum lattice width among all three-dimensional convex bodies. We here prove a local version of this conjecture:…

Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endowed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local…

We construct a convex body $K$ in $\mathbb{R}^n$, $n \geq 5$, with the property that there is exactly one hyperplane $H$ passing through $c(K)$, the centroid of $K$, such that the centroid of $K\cap H$ coincides with $c(K)$. This provides…

Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and…

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…

The Separation Problem asks for the minimum number s(O,K) of hyperplanes required to strictly separate any interior point O of a convex body K from all faces of K. The Conjecture is s(O,K) is at most 2 to the power d in real d-space , and…

We study the geometry of hyperconvex representations of hyperbolic groups in ${\rm PSL}(d,\mathbb{C})$ and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its…

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to…

We study the $l_p$ Hausdorff distance from convex hull of a compact set $A\subset\mathbb{R}^n$, which is the distance \begin{equation*} d^{(l_p)}(A):=\sup_{x\in conv(A)}\inf_{a\in A}\|x-a\|_p, \end{equation*} where $\|\cdot\|_p$ is the…