We study the monoid generated by $$n \times n$$ distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ , and we compute the structure of this fan for $$n$$ up to $$5$$ . The monoid captures gossip among $$n$$ gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.

We explore the minimum energy configurations of repelling particles distributed over $n$ possible locations forming a toric grid. We conjecture that the most energy-efficient way to distribute $n/2$ particles over this space is to place them in a checkerboard pattern. Numerical experiments validate this conjecture for reasonable choices of the repelling force. In the present paper, we prove this conjecture in a large number of special cases---most notably, when the sizes of the torus are either two or multiples of four in all dimensions and the repelling force is a completely monotonic function of the Lee distance between the particles.

We show that the number of lattice directions in which a convex body in $\mathbb{R}^d$ has minimum width is at most $3^d-1$, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the $3^d$-theorem due to Minkowski.