Balázs Csikós

On the Volume of the Union of Balls

Abstract. We prove that if some balls in the Euclidean space move continuously in such a way that the distances between their centers decrease, then the volume of their union cannot increase. The proof is based on a formula expressing the derivative of the volume of the union as a linear combination of the derivatives of the distances between the centers with nonnegative coefficients.

On the Volume of Flowers in Space Forms

Let f(x1,..., xN) be a lattice polynomial in N variables, in which each variable occurs exactly once, B1,..., BN be smoothly moving balls in the hyperbolic, Euclidean, or spherical space. Introducing a suitable modification of the Dirichlet–Voronoi decomposition, we prove a formula for the derivative of the volume of the domain f(B1,..., BN). As an application of the formula, we show that the volume increases if the balls move continuously in such a way that the functions εijdij increase for all 1 ≤ i < j ≤ N, where εij is a sign depending on f, dij is the distance between the centers of BI and Bj.

On the number of triple points of an immersed surface with boundary

Given a generic immersionf:S1→S2 of a circle into the sphere, we find the best possible lower estimation for the number of triple points of a generic immersionF: (M, S1)→(B3,S2) extendingf, whereM is an oriented surface with boundary ∂M=S1,B3 is the 3-dimensional ball with boundaryS2.

Large antipodal families

A family {Ai | i ∈ I} of sets in ℝd is antipodal if for any distinct i, j ∈ I and any p ∈ Ai, q ∈ Aj, there is a linear functional ϕ:ℝd → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪i∈IAi. We study the existence of antipodal families of large finite or infinite sets in ℝ3.

REMARKS ON THE ZIG-ZAG THEOREM

The classical Zig-zag Theorem [1] says that if an equilateral closed 2m-gon shuttles between two given circles of the Euclidean 3-space, then the vertices of the polygon can be moved smoothly along the circles without changing the lengths of the sides of the polygon. First we prove that the Zig-zag Theorem holds also in the hyperbolic, Euclidean and spherical n-spaces, and in fact the circles can be replaced by straight lines or any kind of cycles. In the second part of the paper we restrict our attention to planar zig-zag configurations. With the help of an alternative formulation of the Zig-zag Theorem, we establish two duality theorems for periodic zig-zags between two circles.

Two Kneser–Poulsen-type inequalities in planes of constant curvature

We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase. This generalizes the theorem on the monotonicity of the perimeter of the convex hull of a finite set under contractions, proved in the Euclidean plane by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3]. We also prove that the area of the intersection of finitely many disks in the hyperbolic plane does not decrease after such a contractive rearrangement. The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].

Harmonic Manifolds and Tubes

Csikós and Horváth (J Lond Math Soc (2) 94(1):141–160, 2016) showed that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.

Calgary Workshop in Discrete Geometry (Friday, May 13, 2005 to Saturday, May 14, 2005)

SummaryThis paper is a report on the Calgary Workshop in Discrete Geometry which was held on May 13-14, 2005 at the Department of Mathematics and Statistics of the University of Calgary, Canada. The Workshop was organized to honor the fiftieth birthday of K. Bezdek. Beside a brief description, the report contains the schedule and abstracts of the talks presented at the meeting.

Small convex polytopes with long edges and many vertices

We study how small the diameter of a d-dimensional convex polytope can be if its edges are not shorter than 1 and the number of its vertices is large. We give upper and lower bounds coinciding asymptotically in dimensions d = 3 and d ≥ 7 as the number of vertices tends to infinity.

On the Principal Angles between the Osculating K-Planes of a Curve

For a curve of general type lying in a Euclidean space, we compute the first nonzero term in the Taylor expansion of the principal angles between the osculating k-dimensional subspaces of the curve at the points P0 and P with respect to the length of the arc between P0 and P, whenever the principal angle is not 0 identically by dimension arguments.