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Register a new userKissing number in non-Euclidean spaces of constant sectional curvature
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This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $mathbb{H}^n$ and spherical $mathbb{S}^n$ spaces, for $ngeq 2$. 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 $kappa_H(n,r) sim (n-1) cdot d_{n-1} cdot B(frac{n-1}{2}, frac{1}{2}) cdot e^{(n-1) r}$, where $d_n$ is the sphere packing density in $mathbb{R}^n$, 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.
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This paper investigates the behaviour of the kissing number $kappa(n, r)$ of congruent radius $r > 0$ spheres in $mathbb{S}^n$, for $ngeq 2$. Such a quantity depends on the radius $r$, and we plot the approximate graph of $kappa(n, r)$ with relatively high accuracy by using new upper and lower bounds that are produced via semidefinite programming and by using spherical codes, respectively.
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $mathbb{H}^n$, for $ngeq 2$. For that purpose, the kissing number is replaced by the kissing function $kappa(n, r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $kappa(n, r)$, we study their asymptotic behaviour and show, in particular, that $lim_{rto infty} frac{log kappa(n,r)}{r} = n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.
We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notion of position vector field, which lies at the core of the Minkowski identities.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class $C^2_+$ with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either $d ge 3$, or $d=2$ and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points --- for $S^d$ of radius less than $ pi /2$ --- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.
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