Ball characterizations in spaces of constant curvature

2018 ◽  
Vol 55 (4) ◽  
pp. 421-478
Author(s):  
Jesus Jerónimo-Castro ◽  
Endre Makai, Jr.
Keyword(s):  
Convex Sets ◽  
Central Symmetry ◽  
Connected Components ◽  
Plane Convex ◽  
Converse Implication ◽  
Plane Convex Body ◽  
Dual Theorem ◽  
Centrally Symmetric ◽  
Whole Space ◽  
Straight Lines

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

An extremal property of the hypersphere

1951 ◽  
Vol 47 (1) ◽  
pp. 245-247 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Convex Body ◽  
Extremal Property ◽  
Plane Case ◽  
Simple Application ◽  
Plane Convex ◽  
Plane Convex Body

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

On topological properties of weakly m-convex sets

2021 ◽  
Vol 34 ◽  
pp. 75-84
Author(s):  
Tetiana Osipchuk
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Topological Classification ◽  
Connected Components ◽  
Topological Properties ◽  
Convex Domains ◽  
Closed Set ◽  
Dimensional Plane ◽  
Open Set ◽  
Plane Passing

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.

scholarly journals On the relative distances of seven points in a plane convex body

2007 ◽  
Vol 87 (1-2) ◽  
pp. 83-95 ◽  
Author(s):  
Antal Joós ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Plane Convex ◽  
Plane Convex Body

scholarly journals On the relative distances of six points in a plane convex body

2005 ◽  
Vol 42 (3) ◽  
pp. 253-264
Author(s):  
Károly Böröczky ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Relative Distance ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

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