Related papers: Maximal perimeter and maximal width of a convex sm…
Let $K$ be a convex pentagon in the plane and let $K_1$ be the pentagon bounded by the diagonals of $K$. It has been conjectured that the maximum of the ratio between the areas of $K_1$ and $K$ is reached when $K$ is an affine regular…
We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the…
Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and…
The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the…
Consider the regular $n$-simplex $\Delta_n$ - it is formed by the convex-hull of $n+1$ points in Euclidean space, with each pair of points being in distance exactly one from each other. We prove an exact bound on the width of $\Delta_n$…
Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised…
We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we…
In this paper we are interested in "optimal" universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term "optimal" is to be understood in the following sense: we tackle the issue of…
Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A…
Recently W. Lao and M. Mayer [6], [7], [9] considered $U$-max - statistics, where instead of sum appears the maximum over the same set of indices. Such statistics often appear in stochastic geometry. The examples are given by the largest…
The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic…
This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be…
We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.
Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings…
The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…
Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound…
A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…
We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies,…
This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball…
We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…