Related papers: Shadows of convex bodies

We study geometric properties of coordinate projections. Among other results, we show that if a body K in R^n has an "almost extremal" volume ratio, then it has a projection of proportional dimension which is close to the cube. We compare…

We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Gruenbaum about the "size" of the set of all affine invariant points.…

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where…

We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy,…

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio…

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether…

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…

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…

Every convex polygon with $n$ vertices is a linear projection of a higher-dimensional polytope with at most $147\,n^{2/3}$ facets.

A shadow of a geometric object $A$ in a given direction $v$ is the orthogonal projection of $A$ on the hyperplane orthogonal to $v$. We show that any topological embedding of a circle into Euclidean $d$-space can have at most two shadows…

The quantum scattering by smooth bodies is considered for small and large values of $kd$, with $k$ the wavenumber and $d$ the scale of the body. In both regimes, we prove that the forward scattering exceeds the backscattering. For high $k$,…

In ["Illumination of convex bodies with many symmetries", Mathematika 63 (2017)], Tikhomirov verified the Hadwiger-Boltyanski Illumination Conjecture for the class of 1-symmetric convex bodies of sufficiently large dimension. We propose an…

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.…

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for…

For a fixed $k\in\{1,\dots,d\}$ consider random vectors $X_0,\dots, X_{k}\in\mathbb R^d$ with an arbitrary spherically symmetric joint density function. Let $A$ be any non-singular $d\times d$ matrix. We show that the $k$-dimensional volume…

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:…

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with $L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $\lambda$. J.…

The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterising ellipsoids of maximal volume contained in convex bodies.

In this paper, we introduce an $m$-fold illumination number $I^m(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$, which is the smallest number of directions required to $m$-fold illuminate $K$, i.e., each point on the boundary of…

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability…