Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Covering convex bodies by cylinders and lattice points by flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d

Piercing translates and homothets of a convex body

According to a classical result of Grünbaum, the transversal number τ(F) of any family F of pairwise-intersecting translates or homothets of a convex body C in R d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ(F) to the packing number ν(F) over all families F of translates (resp. homothets) of a convex body C in R d. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in R d, and gave the first bounds on α(C) for convex bodies C in R d and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in R d, and present new or improved bounds on both α(C) and β(C) for various convex bodies C in R d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body