Balancing games on unbounded sets

Abstract

For a finite set $V\subset {\mathbb{R}}^n$, a set $T\subset {\mathbb{R}}^n$ is called $V$-closed if $t\in T$ and $v\in V$ imply that either $t+v\in T$ or $t-v\in T$. The set $P(V):=\{{\sum }_{v\in W}v:W\subset V\}$ is clearly $V$-closed and so are its translates. We show, assuming $V$ contains no parallel vectors, that if $T$ is closed and $V$-closed, and $x\in T$ is an extreme point of $\mathrm{cl}\mathrm{conv}T$, then there is a translate of $P(V)$ containing $x$ and contained in $\mathrm{conv}T$. This result is used to determine the value of a special balancing game. A byproduct is that when $m\ge 2$ and is not a power of 2, then the $m$-sets of a $2m$-set can be coloured Red and Blue so that complementary $m$-sets have distinct colours and every point of the $2m$-set is contained in the same number of Red and Blue sets.

17 pages