On the vertex index of convex bodies

Proceedings of the Steklov Institute of Mathematics, 2011

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 "fat" one, if it contains the centers of its generating balls. The main result of this paper is a proof of the Illumination Conjecture for "fat" spindle convex bodies in dimensions greater than or equal to 15.

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2013

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach [2] asked the following question: For n ≥ 5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine [1] answered this question, and showed that the supremum is the perimeter of an isosceles triangle inscribed in the disk, with an edge of multiplicity n − 2. In [3], Lángi generalized their result for polygons contained in a hyperbolic disk. In this note we find the supremum of the perimeters of simple n-gons contained in an arbitrary plane convex body in the Euclidean or in the hyperbolic plane.

Publicationes Mathematicae Debrecen

Let C ⊂ E 2 be a convex body. The C-length of a segment is the ratio of its length to the half of the length of a longest parallel chord of C. By a relatively equilateral polygon inscribed in C we mean an inscribed convex polygon all of whose sides are of equal C-length. We prove that for every boundary point x of C and every integer k ≥ 3 there exists a relatively equilateral k-gon with vertex x inscribed in C. We discuss the C-length of sides of relatively equilateral k-gons inscribed in C and we reformulate this question in terms of packing C by k homothetical copies which touch the boundary of C. Let C be a convex body in Euclidean n-space E n. If pq is a longest chord of C in a direction l, we say that points p and q are opposite and we call pq a diametral chord of C in direction l. By the C-distance dist C (a, b) of a and b we mean the ratio of the Euclidean distance |ab| of a and b to the half of the Euclidean distance of end-points of a diametral chord of C parallel to ab (comp. [7]). We use here the term relative distance if there is no doubt about C. By the C-length of the segment ab we mean dist C (a, b). If C ⊂ E 2 , we define a C-equilateral k-gon as a convex k-gon all of whose sides have equal C-lengths. We also use the name relatively equilateral k-gon when C is fixed. Section 1 is of an auxiliary nature. It presents properties of the Cdistance, and especially properties of the C-distance of boundary points

Canadian Mathematical Bulletin, 2009

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ℝ d is 2 d . Naszódi proved that the quantity in question is not larger than 2 d+1. We present an improvement to this result by proving the upper bound 3 · 2 d–1 for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

Geometriae Dedicata, 1986

According to the context, in the following two properties we assume that PlffP2 and qlJ(q2. (i) Any three conditions of (*) imply the fourth one. (ii) If pl, P2 and ql, q2 are pairs of vectors fulfilling (*), then IPll/lql-q21 =lP21/Iqx +q2[, ]q~I/IPl +-P2I-lq2l/IPx-PEI, and pq = ½, where p = IPll/Iql-qEI and q-Iq~l/IPl + P21.