Related papers: Constant diameter and constant width of spherical …

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

In this paper we present a unique 4-dimensional body of constant width based on the classical notion of focal conics.

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…

The purpose of this article is to give an explicit formula for all curves of constant torsion $\tau$ in the unit two-sphere $S^2(1)$. These curves and their basic properties have been known since the 1890's, and some of these properties are…

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of…

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…

The study of bodies of constant width is a classical subject in convex geometry, with the 3-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $\mathbb…

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball…

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the…

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a…

It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies ls are studied.

We show stability of spherical caps (SCs) lying on a flat surface, where the motion is governed by the volume-preserving Mean Curvature Flow (MCF). Moreover, we introduce a dynamic boundary condition that models a line tension effect on the…

Let $\gamma: S^n\to \mathbb{R}_+$ be a convex integrand and $\mathcal{W}_\gamma$ be the Wulff shape of $ \gamma$. Apex point naturally arise in non-smooth Wulff shape, in particular, vertex of convex polytope. %Let $P\in S^n$. In this…

In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure…

If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.

A circle C holds a convex body K if C does not meet the interior of K and if there does not exist any euclidean displacement which moves C as far as desired from K, avoiding the interior of K. The purpose of this note is to explore how…

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls…

Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $C^1$. Moreover, applications of this result are given.