JournalsjemsVol. 15, No. 6pp. 2297–2352

Operations between sets in geometry

  • Richard J. Gardner

    Western Washington University, Bellingham, USA
  • Daniel Hug

    Karlsruher Institut für Technologie (KIT), Germany
  • Wolfgang Weil

    Karlsruher Institut für Technologie (KIT), Germany
Operations between sets in geometry cover
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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 nn-dimensional Euclidean space Rn\mathbb R^n. It is proved that if n2n\ge 2, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n)GL(n) covariant, and associative if and only if it is LpL_p addition for some 1p1\le p\le\infty. It is also demonstrated that if n2n\ge 2, an operation * between compact convex sets is continuous in the Hausdorff metric, GL(n)GL(n) covariant, and has the identity property (i.e., K{o}=K={o}KK*\{o\}=K=\{o\}*K for all compact convex sets KK, where oo 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 MM-addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and GL(n)GL(n)-covariant operations between compact convex sets in terms of MM-addition are established. It is shown that if n2n\ge 2, an oo-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, GL(n)GL(n) covariant, and translation invariant if and only if it is of the form λDK\lambda DK for some λ0\lambda\ge 0, where DK=K+(K)DK=K+(-K) is the difference body of KK. The term ``polynomial volume" is introduced for the property of operations * between compact convex or star sets that the volume of rKsLrK*sL, r,s0r,s\ge 0, is a polynomial in the variables rr and ss. It is proved that if n2n\ge 2, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, GL(n)GL(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition.

Cite this article

Richard J. Gardner, Daniel Hug, Wolfgang Weil, Operations between sets in geometry. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2297–2352

DOI 10.4171/JEMS/422