16 papers · 1 filter
Covering a square by congruent squares
György Dósa, Zsolt Lángi, Zsolt Tuza
The main goal of this paper is to address the following problem: given a positive integer , find the largest value such that a square of edge length in the Euclide…
Non-separable arrangements of convex bodies revisited
Károly Bezdek, Zsolt Lángi
A problem posed by Erdős in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean -space is called a non-sep…
Curves with increasing chords in normed planes
Zsolt Lángi, Sára Lengyel
A curve has the increasing chord property if for any points in this order on the curve, the distance of is not smaller than that of . Answering a conjecture of…
John Ellipsoids of Revolution
Grigory Ivanov, Zsolt Lángi, Márton Naszódi +1
Finding a largest Euclidean ball in a given convex body and finding a largest volume ellipsoid in are two problems of fundamentally different nature. T…
Selected topics from the theory of intersections of balls
Károly Bezdek, Zsolt Lángi, Márton Naszódi
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the -dimensional real vect…
On separability in discrete geometry
Károly Bezdek, Zsolt Lángi
A problem of Erdős (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes Tóth and Fejes Tóth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-s…
Centroids and equilibrium points of convex bodies
Zsolt Lángi, Péter L. Várkonyi
The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect resu…
The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker
Zsolt Lángi, Shanshan Wang
The Honeycomb Conjecture states that among tilings with unit area cells in the Euclidean plane, the average perimeter of a cell is minimal for a regular hexagonal tiling. This conj…
Steiner symmetrization on the sphere
Bushra Basit, Steven Hoehner, Zsolt Lángi +1
The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbol…
Density bounds for unit ball packings relative to their outer parallel domains
Károly Bezdek, Zsolt Lángi
We prove that the highest density of non-overlapping translates of a given centrally symmetric convex domain relative to its outer parallel domain of given outer radius is attained…
On the edge densities of normal, convex mosaics
Máté Kadlicskó, Zsolt Lángi, Shanxiang Lyu
In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in -dimensional Euclidean space. In the…
Remarks on soft ball packings in dimensions 2 and 3
Károly Bezdek, Zsolt Lángi
We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean -space) that neither pack nor cover. We define t…
On a Dowker-type problem for convex disks with almost constant curvature
Bushra Basit, Zsolt Lángi
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body , the areas of the maximum (resp. minimum) area convex -gons ins…
Dowker-type theorems for disk-polygons in normed planes
Bushra Basit, Zsolt Lángi
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body in the Euclidean plane, the areas of the maximum (resp. minimum) a…
On optimal -separable packings in the plane
Károly Bezdek, Zsolt Lángi
Let be a packing of circular disks of radius in the Euclidean, spherical, or hyperbolic plane. Let . We say that is a -separable p…
Hexagon tilings of the plane that are not edge-to-edge
Dirk Frettlöh, Alexey Glazyrin, Zsolt Lángi
An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer , th…