Operations between sets in geometry

Abstract

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space ${\mathbb{R}}^n$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $\mathrm{GL}(n)$ covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le \infty$. It is also demonstrated that if $n\ge 2$, an operation $\ast$ between compact convex sets is continuous in the Hausdorff metric, $\mathrm{GL}(n)$ covariant, and has the identity property (i.e., $K\ast \{o\}=K=\{o\}\ast K$ for all compact convex sets $K$, where $o$ denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. Various characterizations are given of operations that are projection covariant, meaning that the operation can take place before or after projection onto subspaces, with the same effect.

Several other new lines of investigation are followed. A relatively little-known but seminal operation called $M$-addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and $\mathrm{GL}(n)$-covariant operations between compact convex sets in terms of $M$-addition are established. It is shown that if $n\ge 2$, an $o$-symmetrization of compact convex sets (i.e., a map from the compact convex sets to the origin-symmetric compact convex sets) is continuous in the Hausdorff metric, $\mathrm{GL}(n)$ covariant, and translation invariant if and only if it is of the form $\lambda DK$ for some $\lambda \ge 0$, where $DK=K+(-K)$ is the difference body of $K$. The term “polynomial volume” is introduced for the property of operations $\ast$ between compact convex or star sets that the volume of $rK\ast sL$, $r,s\ge 0$, is a polynomial in the variables $r$ and $s$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $\mathrm{GL}(n)$ covariant, associative, and has polynomial volume if and only if it is Minkowski addition.