Mathematics

In this note, we investigate an infinite one parameter family of circle packings, each with a set of three mutually tangent circles. We use these to generate an infinite set of circle packings with the Apollonian property. That is, every circle in the packing is a member of a cluster of four mutually tangent circles.

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree n are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.

If a convex body K⊂ R n is covered by the union of convex bodies C 1 ,…, C N , multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich K ) and the inradius (the largest radius of a ball contained in K ): the sum of the widths of the C i is at least the width of K (this is the plank theorem of Thoger Bang), and the sum of the inradii of the C i is at least the inradius of K (this is due to Vladimir Kadets). We adapt the existing proofs of these results to prove a theorem on coverings by certain generalized non-convex "multi-planks". One corollary of this approach is a family of inequalities interpolating between Bang's theorem and Kadets's theorem. Other corollaries include results reminiscent of the Davenport--Alexander problem, such as the following: if an m -slice pizza cutter (that is, the union of m equiangular rays in the plane with the same endpoint) in applied N times to the unit disk, then there will be a piece of the partition of inradius at least sinπ/m N+sinπ/m .

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in R d given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of ``containing k colinear integer points.

The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257-284, 2005) states that the smallest area convex domain of constant width w in the 2 -dimensional spherical space S 2 is the spherical Reuleaux triangle for all 0<w??? 2 . In this paper we extend this result to the family of wide r -disk domains of S 2 , where 0<r??? 2 . Here a wide r -disk domain is an intersection of spherical disks of radius r with centers contained in their intersection. This gives a new and short proof for the Blaschke-Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide r -disk domains called wide r -ball bodies. In particular, we determine their minimum spherical width (resp., inradius) in the spherical d -space S d for all d?? . Also, it is shown that any minimum volume wide r -ball body is of constant width r in S d , d?? .

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.

In this research-expository paper we recall the basic results of reduction theory of positive definite quadratic forms. Using the result of Ryskov on admissible centerings and the result of Tammela about the determination of a Minkowski-reduced form, we prove that the absolute values of coordinates of a minimum vector in a six-dimensional Minkowski-reduced basis are less or equal to three. To get this little sharpening of the result which can be deduced automatically from Tammela's works we combine some elementary geometric reasonings with the mentioned theoretical results.

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures when their parameter of concavity is positive.

We establish universality and ultra-homogeneity of (U, u GH ) , the collection of all compact ultrametric spaces endowed with the so-called Gromov-Hausdorff ultrametric. This result also gives rise to a novel construction of the so-called R -Uryoshn universal ultrametric space for each countable subset R⊂ R ≥0 containing 0 .

A finite subset of a Euclidean space is called an s -distance set if there exist exactly s values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in R 3 is 20, and every 5 -distance set in R 3 with 20 points is similar to the vertex set of a regular dodecahedron.