Covering a convex body by its negative homothetic copies

Covering a Convex Body by Its Negative Homothetic Copies

Translative Packing of a Convex Body by Sequences of Its Homothetic Copies

Every sequence of positive or negative homothetic copies of a planar convex body C whose total area does not exceed 0.175 times the area of C can be translatively packed in C. Let C be a planar convex body with area |C|. Moreover, let (Ci) be a finite or infinite sequence of homothetic copies of C. We say that (Ci) can be translatively packed in C if there exist translations σi such that σiCi a...

Randomized coverings of a convex body with its homothetic copies, and illumination

We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers’ upper bounds for the illumination number, we improve previous estimates of J. Januszewski and M. Naszódi for a generalized version of the illumination parameter.

The Volume of the Convex Hull of a Body and its Homothetic Copies

In this note, we prove the following result. Let K ⊂ Rd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λK , by a vector x , depends only on the Euclidean norm of x , then K is a Euclidean ball.

Covering the Plane with Congruent Copies of a Convex Body

It is shown that every plane compact convex set /f with an interior point admits a covering of the plane with density smaller than or equal to 8(2\/3 — 3)/3 = 1.2376043 For comparison, the thinnest covering of the plane with congruent circles is of density 2n/\Z21 = 1.209199576... (see R. Kershner [3]), which shows that the covering density bound obtained here is close to the best possible. It ...