arXiv subjects

Zsolt Langi

Publications and source records attributed to Zsolt Langi.

13 recordsLinked to original sources

arXiv2022

Discrete isoperimetric problems in spaces of constant curvature

The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the hyperbolic polytopes with $d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any $1 \leq k \leq d$, we investigate the properties of Euclidean simplices and polytopes with $d+2$ vertices having a fixed inradius and a minimal volume of its $k$-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.

math.MG↗

arXiv2018

The evolution of geophysical shape descriptors under distance-driven flows

We investigate the evolution of axis ratios, roundness (isoperimetric ratio) and the number of static balance points under distance-driven flows. The latter have already been proposed by Aristotle as models of particle shape evolution and recent studies indicate that they may serve as models for frictional abrasion. We show exact conditions under which Aristotle's original claims are true. For several geophysical shape descriptors we prove monotonic or quasiconcave time evolution and compare these results with results from the literature on curvature-driven flows as models of collisional abrasion.

math.MG↗

arXiv2018

A characterization of affinely regular polygons

In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $λ\geq 0$ such that $p_{j+2}-p_{j-1} = λ(p_{j+1}-p_j)$ for $j=1,2,\ldots, n$. The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\ldots,p_n \in \mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \in \mathbb{C}$ and $m_1,m_2,k \in \mathbb{Z}$. In particular, we show that in `most' cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups.

math.MG↗

arXiv2016

On non-separable families of positive homothetic convex bodies

A finite family ${\mathcal B}$ of balls with respect to an arbitrary norm in ${\mathbb R}^d$ ($d\geq 2$) is called a non-separable family if there is no hyperplane disjoint from $\bigcup {\mathcal B}$ that strictly separates some elements of ${\mathcal B}$ from all the other elements of ${\mathcal B}$ in ${\mathbb R}^d$. In this paper we prove that if ${\mathcal B}$ is a non-separable family of balls of radii $r_1, r_2,\ldots , r_n$ ($n\geq 2$) with respect to an arbitrary norm in ${\mathbb R}^d$ ($d\geq 2$), then $\bigcup {\mathcal B}$ can be covered by a ball of radius $\sum_{i=1}^n r_i$. This was conjectured by Erdos for the Euclidean norm and was proved for that case by A. W. Goodman and R. E. Goodman [Amer. Math. Monthly 52 (1945), 494-498]. On the other hand, in the same paper A. W. Goodman and R. E. Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in ${\mathbb R}^d$, $d\ge2$. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.

math.MG↗

arXiv2014

Density bounds for outer parallel domains of unit ball packings

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.

math.MG↗

arXiv2014

On global equilibria of finely discretized curves and surfaces

In an earlier work we identified the types and numbers of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We showed that such discretizations carry equilibrium points on two scales: the local scale corresponds to the discretization, the global scale to the original, smooth surface. In that paper we showed that as $n$ approaches infinity, the number of local equilibria fluctuate around specific values which we call the imaginary equilibrium indices associated with the approximated smooth surface. Here we show how the number of global equilibria can be interpreted, defined and computed on such discretizations. Our results are relevant from the point of view of natural pebble surfaces, they admit a comparison between field data based on hand measurements and laboratory data based on 3D scans.

math.DG↗

arXiv2011

On the equilibria of finely discretized curves and surfaces

Our goal is to identify the type and number of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We assume uniform gravity and a frictionless, horizontal, planar support. We show that as $n$ approaches infinity these numbers fluctuate around specific values which we call the imaginary equilibrium indices associated with the approximated smooth surface. We derive simple formulae for these numbers in terms of the principal curvatures and the radial distances of the equilibrium points of the solid from its center of gravity. Our results are illustrated on a discretized ellipsoid and match well the observations on natural pebble surfaces.

math.DG↗

arXiv2011

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

math.MG↗

arXiv2011

On the Hadwiger numbers of starlike disks

The Hadwiger number $H(J)$ of a topological disk $J$ in $\Re^2$ is the maximal number of pairwise nonoverlapping translates of $J$ that touch $J$. It is well known that for a convex disk, this number is six or eight. A conjecture of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike disk is at most eight. A. Bezdek proved that this number is at most seventy five for any starlike disk. In this note, we prove that the Hadwiger number of a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger number of a topological disk $J$ such that $(\conv J) \setminus J$ is connected, is six or eight.

math.MG↗

arXiv2010

On universal covers for four-dimensional sets of a given diameter

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polytope, with at most fourteen facets and circumscribed about the Euclidean ball of diameter one, is a universal cover for the family of unit diameter sets.

math.MG↗

arXiv2010

On diagonalizable operators in Minkowski spaces with the Lipschitz property

A real semi-inner-product space is a real vector space $\M$ equipped with a function $[.,.] : \M \times \M \to \Re$ which is linear in its first variable, strictly positive and satisfies the Schwartz inequality. It is well-known that the function $||x|| = \sqrt{[x,x]}$ defines a norm on $\M$. and vica versa, for every norm on $X$ there is a semi-inner-product satisfying this equality. A linear operator $A$ on $\M$ is called \emph{adjoint abelian with respect to $[.,.]$}, if it satisfies $[Ax,y]=[x,Ay]$ for every $x,y \in \M$. The aim of this paper is to characterize the diagonalizable adjoint abelian operators in finite dimensional real semi-inner-product spaces satisfying a certain smoothness condition.

math.FA↗

arXiv2010

On the Borsuk number of four-dimensional sets

Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one.

math.MG↗

arXiv2010

On the perimeters of simple polygons contained in a disk

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.

math.MG↗