Related papers: On bodies with congruent sections or projections

Let $K$ and $L$ be two convex bodies in ${\mathbb R^4}$, such that their projections onto all $3$-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfy an additional condition and some…

Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many diameters, such that their projections onto all $4$ dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections…

We prove that if two convex bodies $ K, L \subset \mathbb{R}^3$ satisfy the property that the orthogonal projections of $K$ and $L$ onto every plane containing the origin are roations of each other, then either $K$ and $L$ coincide or $L$…

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…

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M…

Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. In this paper we show that $P$ and $Q$ or $P$ and $-Q$…

In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|\theta^{\perp}|…

Hedgehogs are geometrical objects that describe the Minkowski differences of arbitrary convex bodies in the Euclidean space $\mathbb{E}^n$. We prove that two hedgehogs in $\mathbb{E}^n, n \geq 3$, coincide up to a translation and a…

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies…

In this work we present a theorem regarding two convex bodies $K_1, K_2\subset \mathbb{R}^{n}$, $n\geq 3$, and two families of sections of them, given by two families of tangent planes of two spheres $S_i\subset \textrm{int}\textrm{ } K_i$,…

For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in…

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function…

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…

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…

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 every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare $(n-2)$-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions.

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are…

We study the following open problem, suggested by Barker and Larman. Let $K$ and $L$ be convex bodies in $\mathbb R^n$ ($n\ge 2$) that contain a Euclidean ball $B$ in their interiors. If $\mathrm{vol}_{n-1}(K\cap H) =…

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <=…