Packing a convex domain with similar convex domains

Convex-Bodies with Similar Projections

By examining an example constructed by Petty and McKinney, we show that there are pairs of centered and coaxial bodies of revolution in Ed, d > 3, whose projections onto each two-dimensional subspace are similar, but which are not themselves even affinely equivalent.

Finite packing and covering by congruent convex domains

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in R n has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

Domains with Convex Hyperbolic Radius

The hyperbolic radius of a domain on the Riemann sphere is equal to the reciprocal of the density of the hyperbolic metric. In the present paper, it is proved that the hyperbolic radius is a convex function if and only if the complement of the domain is a convex set.

Packing Convex Bodies by Cylinders

In [BL] in relation to the unsolved Bang’s plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer’s results ([Fa]).