The Convex Hull of a Typical Compact Set.
The convexity of the subset space of a metric space
The Countability of a Tiling Family and the Periodicity of a Tiling.
The critical determinant of the double paraboloid and Diophantine approximation in and .
The Cubical d-Polytopes with Fewer than 2d+1 Vertices.
The cyclic sieving phenomenon for faces of cyclic polytopes.
The Demyanov metric and some other metrics in the family of convex sets
We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.
The densest geodesic ball packing by a type of Nil lattices.
The densest lattice in twenty-four dimensions.
The densest packing of 12 congruent circles in a circle.
The densest packing of 13 congruent circles in a circle.
The Densest Packing of Equal Circles into a Parallel Strip.
The determination of convex bodies from the size and shape of their projections and sections
We survey results concerning the extent to which information about a convex body's projections or sections determine that body. We will see that, if the body is known to be centrally symmetric, then it is determined by the size of its projections. However, without the symmetry condition, knowledge of the average shape of projections or sections often determines the body. Rather surprisingly, the dimension of the projections or sections plays a key role and exceptional cases do occur but appear to...
The Different Ways of Stabbing Disjoint Convex Sets.
The dimension of almost spherical sections of convex bodies
The Dimension Print of Most Convex Surfaces.
The Disk-Packing Constant
The dissection of five-dimensional simplices into orthoschemes.
The distance between subdifferentials in the terms of functions
For convex continuous functions defined respectively in neighborhoods of points in a normed linear space, a formula for the distance between and in terms of (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly...
The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations
A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented...