Balancing games on unbounded sets
Imre Bárány, Jeck Lim
TL;DR
The paper investigates V-closed sets and their relationship to balancing games in Euclidean space, proving that under a no-parallel-vectors condition, every extreme point of the convex hull of a V-closed set lies in a translate of P(V) contained in the hull. This geometric insight yields exact thresholds for a two-player balancing game G(V,K_M), showing exponential-in-n behavior and enabling a sharp combinatorial corollary: a Red/Blue coloring of m-subsets of a 2m-set with prescribed coverage and complementary-set distinctions. The authors combine planar reductions, projection techniques, and constructive sign assignments to control translates of P(V) and to derive both game-theoretic and purely combinatorial consequences. The results illuminate the interplay between discrete geometry and combinatorial coloring, offering precise thresholds and optimality statements depending on parity and powers of two.
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.