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 $\operatorname{cl} \operatorname{conv} T$, then there is a translate of $P(V)$ containing $x$ and contained in $\operatorname{conv} T$. 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.