We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.

We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan, the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.

We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof). 

The average kissing number in ℝn is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in ℝn. We provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions 3,.., 9. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions 6,.., 9 our new bound is the first to improve on this simple upper bound.

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. This algorithm does not require the solution to be strictly feasible and works for large problems. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the E-8 root lattice is the unique optimal code with minimal angular distance pi/3 on the hemisphere in R-8, and we prove that the three-point bound for the (3, 8, theta)-spherical code, where theta is such that cos theta = (2 root 2-1/7, is sharp by rounding to Q[root 2]. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.

This paper provides upper and lower bounds on the kissing number of congruent radius r > 0 spheres in hyperbolic H-n and spherical S-n spaces, for n >= 2. For that purpose, the kissing number is replaced by the kissing function kappa(H)(n, r), resp. kappa(S)(n, r), which depends on the dimension n and the radius r. After we obtain some theoretical upper and lower bounds for kappa(H)(n, r), we study their asymptotic behaviour and show, in particular, that kappa(H)(n, r) similar to (n - 1) . d(n-1) . B(n-1/2, 1/2) . e((n-1)r), where d(n) is the sphere packing density in Rn, and B is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of kappa(S)(n, r), for n = 3, 4, over subintervals in [0, pi] with relatively high accuracy.

This paper provides upper and lower bounds on the kissing number of congruent radius r> 0 spheres in hyperbolic Hn and spherical Sn spaces, for n≥ 2. For that purpose, the kissing number is replaced by the kissing function κH(n, r), resp. κS(n, r), which depends on the dimension n and the radius r. After we obtain some theoretical upper and lower bounds for κH(n, r), we study their asymptotic behaviour and show, in particular, that κH(n,r)∼(n-1)·dn-1·B(n-12,12)·e(n-1)r, where dn is the sphere packing density in Rn, and B is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of κS(n, r), for n=3,4, over subintervals in [ 0, π] with relatively high accuracy.