On classes of convex sets that permit plane coverings

A family {K _i | of convex domains in the Euclidean planeR ² is said to permit a plane covering if there exist rigid motions {τ_i} such that U _i ^∞ =₁τ_i K _i =R ². Necessary and sufficient conditions that a given family of convex domains permits a plane covering are established.     

Homothetic packings of centrally symmetric convex bodies

Geometriae Dedicata - We estimate the bottom of the $$L^2$$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main...     

Mean projections and finite packings of convex bodies

Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inE ^d ,n be large. IfQ has minimali-dimensional projection, 1i<d then we prove thatQ is approximately a sphere.     

Double-lattice packings of convex bodies in the plane

Mahler [7] and Fejes Tóth [2] proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least 3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Tóth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of–K. The lower bound of 3/2 for packing densities produced here is an improvement of the bounds obtained previously in [5] and [6].     

On reduced convex bodies

A convex bodyK ⊂R ^d is called reduced if for each convex bodyK′ ⊂K,K′ ≠K, the width ofK′ is less than the width ofK. We prove that reduced bodyK is of constant width if (i) the bodyK has a supporting sphere almost everywhere in ∂K. (The radius of the sphere may vary with the point in ∂K; the condition (i) and strict convexity do not imply each other.) Supported by an N.S.E.R.C. Grant of Canada.     

On asymmetry of some convex bodies

We determine Minkowski's measure of asymmetry for convex hulls of a point and some sets for which the asymmetry is known. Some properties of the asymmetry measure are found which may indicate some interesting properties of convex bodies. Received March 1, 2001, and in revised form June 29, 2001. Online publication December 17, 2001.     

On convex bodies of constant width

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

On inner quermasses of convex bodies

The author wishes to thank Professor P. Schneider and E. Lutwak for helpful hints.     

On steiner points of convex bodies

Ifs is a mapping from the set of all convex bodies in Euclidean spaceE ^d toE ^d which is additive (in the sense of Minkowski), equivariant with respect to proper motions, and continuous, thens(K) is the Steiner point of the convex bodyK.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

On convex bodies close to ellipsoids

It is known that the ellipsoids ind-dimensional Euclidean space, ford 3, are characterized among all convex bodies by the property that the intersection points of anyd pairwise orthogonal supporting hyperplanes lie on a fixed sphere. The subject of the present note is a quantitative improvement of this uniqueness theorem in the form of a stability result.Herrn Oswald Giering zum 60. Geburtstag gewidmet     

On the illumination of smooth convex bodies

The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213.