Operations between sets in geometry
Richard J. Gardner, Daniel Hug, and Wolfgang Weil

TL;DR
This paper characterizes operations on convex and star sets in Euclidean space, showing that under certain natural conditions, these operations are essentially Minkowski or $L_p$ additions, and introduces the concept of polynomial volume.
Contribution
It provides a complete characterization of set operations with natural properties, introduces $M$-addition, and studies polynomial volume operations systematically.
Findings
Minkowski addition uniquely characterized by continuity, covariance, and identity property.
$L_p$ addition characterized by continuity, covariance, and associativity.
Polynomial volume operations are essentially Minkowski addition.
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 -dimensional Euclidean space . For example, it is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and associative if and only if it is addition for some . It is also demonstrated that if , an operation * between compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and has the identity property (i.e., for all compact convex sets , where denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. An…
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Taxonomy
TopicsPoint processes and geometric inequalities · Mathematics and Applications · Advanced Differential Geometry Research
