Marek Lassak

Reduced convex bodies in the plane

A convex bodyR of Euclideand-spaceEd is called reduced if there is no convex body properly contained inR of thickness equal to the thickness Δ(R) ofR. The paper presents basic properties of reduced bodies inE2. Particularly, it is shown that the diameter of a reduced bodyR⊂E2 is not greater than √2Δ(R), and that the perimeter is at most (2+½π)Δ(R). Both the estimates are the best possible.

On-Line Packing Sequences of Cubes in the Unit Cube

AbstractAlmost thirty years ago Meir and Moser proved that every sequence of d -dimensional cubes of total volume 2

Estimates for the minimal width of polytopes inscribed in convex bodies

\frac{1}{2}

On relatively short and long sides of convex pentagons

)d can be packed in the unit cube. We show that if d≥ 5, then this property holds true also for the on-line packing.

Covering the boundary of a convex set by tiles

For every plane convex body there is a pair of inscribed and circumscribed homothetic rectangles. The positive ratio of homothety is not greater than 2.

Approximation of Convex Bodies by Centrally Symmetric Bodies

The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular, ℋd denote the family of all convex bodies in Euclideand-space, letA ⊂ ℋd and letn be an integer greater thand. Then we ask for the greatest number μ=Λn(A) such that everyA εA contains a polytope withn vertices which has minimal width at least μw(A). We give bounds for Λn(ℋd), for Λn(ℳ2133;d), and for Λn(Wd), where ℳ2133;d,Wd denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.

Approximation of convex bodies by triangles

Abstract. It is proved that the maximum possible volume of a parallelotope contained in a d -dimensional simplex S is equal to (d! / dd) vol(S) . A description of all the parallelotopes of maximum volume contained in S is given.

An on-line potato-sack theorem

By the relative distance of pointsa andb of a convex bodyC we mean the ratio of the Euclidean distance ofa andb to the half of the Euclidean distance ofa′, b′ ∈C such thata′b′ is a longest chord ofC parallel to the segmentab. We say that a sideab of a convexn-gon is relatively short (respectively: relatively long) if the relative distance ofa andb is at most (respectively: at least) the relative distance of two consecutive vertices of the regularn-gon. We show that every convexn-gon, wheren ≤ 5, has a relatively short side and a relatively long side, and that it is affine-regular if and only if all its sides are of equal relative lengths.