Mathematics

Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland and Youssef. This, together with an appropriate selection in the accuracy of the approximation, allow us to obtain Helly-type versions which are sensitive to the number of convex sets involved.

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere S n−1 . After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.

We consider substitution tilings in R^d that give rise to point sets that are not bounded displacement (BD) equivalent to a lattice and study the cardinality of BD(X), the set of distinct BD class representatives in the corresponding tiling space X. We prove a sufficient condition under which the tiling space contains continuously many distinct BD classes and present such an example in the plane. In particular, we show here for the first time that this cardinality can be greater than one.

We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the convergence of the original sequence from that of the transformed sequence, respectively. As an application, we prove that spheres and projective spaces with standard Riemannian distance converge to a Gaussian space and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges to infinity.

Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of i -dimensional subspaces of R n . Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions i of the subspaces, are found by combining two groups of new results. In the first, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set F of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in F appearing infinitely often in the sequence. The second group of results provides finite sets F of subspaces such that symmetry with respect to each subspace in F implies full rotational symmetry. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of i -dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets.

Abstract In this paper, we study affine bodies of revolution. This will allow us to prove that a convex body all whose orthogonal n -projections are affinely equivalent is an ellipsoid, provided n≡0,1,2 mod 4 , n>1 with the possible exemption of n=133 . Our proof uses convex geometry and topology of compact Lie groups. AMS classification subject: 22E10, 52A05

We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.

We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the standard inner product, the Legendre transform associated with the given norm. This approach yields re-interpretations of various properties of convex functions, and new relations between such functions and geometric properties of the studied norm are also derived.

In this study, the properties of convex pentagons that can generate rotationally symmetric edge-to-edge tilings are discussed. Since the rotationally symmetric tilings are formed by concave octagons that are generated by two convex pentagons connected through a line symmetry, they are considered to be equivalent to rotationally symmetric tilings with concave octagons. In addition, under certain circumstances, tiling-like patterns with a regular polygonal hole at the center can be formed using these convex pentagons.

In this study, the properties of convex hexagons that can generate rotationally symmetric edge-to-edge tilings are discussed. Since the convex hexagons are equilateral convex parallelohexagons, convex pentagons generated by bisecting the hexagons can form rotationally symmetric non-edge-to-edge tilings. In addition, under certain circumstances, tiling-like patterns with an equilateral convex polygonal hole at the center can be formed using these convex hexagons or pentagons.