Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps
European Journal of Combinatorics
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What is the minimum radius $\rho_d(n)$ of a spherical container in ${\mathbb R}^d (d\ge 2)$ that can hold n unit balls, and how must the balls be arranged in such a container? This question is equivalent to: How should n points be selected in the unit ball in ${\mathbb R}^d$ so that the minimum distance between any two of them be as large as possible, and what is that distance? Davenport and Hajos, and, independently, Rankin, proved that if F is a set of d + 2 points in the unit ball in ${\mathbb R}^d$, then two of the points in F are at a distance of at most $\sqrt 2$ from each other. Rankin proved also that if F consists of 2d points in the ball such that the distance between any two of them is at least $\sqrt 2$, then their configuration is unique up to an isometry, namely the points must be the vertices of a regular d-dimensional crosspolytope inscribed in the ball. However, if d + 2 ≤ n ≤ 2d - 1, then the optimal arrangements of n points (i.e., those that maximize the smallest distance between them) are not unique. Here we generalize the results from Davenport and Hajos and Rankin by describing all possible optimal configurations, unique or not, of n = d + 2, d + 3,..., 2d points.