Mathematics

Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Molnár et al. in 2006 . As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~Tóth for balls and horoballs in 1964 .

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {\em average} radius of the balls, resulting in bounds of the familiar classical form.

We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.

The structure of k -diametral point configurations in Minkowski d -space is shown to be closely related to the properties of k -antipodal point configurations in R d . In particular, the maximum size of k -diametral point configurations of Minkowski d -spaces is obtained for given k≥2 and d≥2 generalizing Petty's results (Proc. Am. Math. Soc. 29: 369-374, 1971) on equilateral sets in Minkowski spaces. Furthermore, bounds are derived for the maximum size of k -diametral point configurations in Euclidean d -space. In the proofs convexity methods are combined with volumetric estimates and combinatorial properties of diameter graphs.

We show that every L -BLD-mapping in a domain of R n is a local homeomorphism if L< 2 – √ or K I (f)<2 . These bounds are sharp as shown by a winding map.

We study geometric and topological properties of singularities on the boundaries of ε -neighbourhoods E ε ={x??R 2 :dist(x,E)?�ε} of planar sets E??R 2 . We develop a novel technique for analysing the boundary and obtain, for a compact set E and ε>0 , a classification of singularities (i.e. non-smooth points) on ??E ε into eight categories. We also show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set.

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to Ball to the context of q− concave, 1 q − homogeneous measures.

Choose N unoriented lines through the origin of R d+1 . The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for R d+1 . For d≥2 we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the α -th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the {\em unique} optimizer (up to rotations) for all α>1 . We establish both the asserted optimality and uniqueness in the limiting case α=∞ of mildest repulsion. The same conclusions extend to N=∞ , provided we assume only finitely many of the lines are distinct.

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and the hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for the Alesker algebra is conjectured, and the conjecture is proved in two particular situations: for even valuations, and for 1-homogeneous valuations. The latter result is then used to deduce a special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lefschetz theorem and of the Hodge-Riemann relations are conjectured, and it is shown that the Aleksandrov-Fenchel inequality follows from the latter in its full generality.

In this paper, the functional Quermassintegrals of log-concave functions in R n are discussed, we obtain the integral expression of the i -th functional mixed Quermassintegrals, which are similar to the integral expression of the i -th Quermassintegrals of convex bodies.