A Lower Bound for the Translative Kissing Numbers of Simplices

First we recall some standard definitions. By a d-dimensional convex body we mean a compact convex subset of Rd with non-empty interior. Two subsets of Rd with non-empty interiors are non-overlapping if they have no common interior point, and we say that they touch each other if they are non-overlapping and their intersection is non-empty. Denote by H(K) the translative kissing number of a d-dimensional convex body K, which is defined as the maximum number of mutually non-overlapping translates of K that can be arranged so that all touch K. H(K) is often called the Hadwiger number of K as well. By a result of Swinnerton-Dyer [18] it follows that H(K)≥d2+d holds for every d-dimensional convex body K (d≥1). Recently, Talata [19] improved on this lower bound for sufficiently large values of d, showing that there exists an absolute constant c > 0 such that H(K) ≥ 2cd for every d-dimensional