Papers › Kissing number in non-Euclidean spaces of constant sectional curvature
Kissing number in non-Euclidean spaces of constant sectional curvature
Maria Dostert, Alexander Kolpakov
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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 κ_H(n, r), resp. κ_S(n, r), which depends on the dimension n and the radius r. After we obtain some theoretical upper and lower bounds for κ_H(n, r), we study their asymptotic behaviour and show, in particular, that κ_H(n,r) ∼(n-1) ·dₙ₋₁ ·B((n-1)/2, 1/2) ·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 κ_S(n, r), for n= 3, 4, over subintervals in [0, π] with relatively high accuracy.
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