Geodesic
Operations between sets in geometry
Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2297-2352

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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 Rn. It is proved that if n≥2, 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 Lp​ addition for some 1≤p≤∞. It is also demonstrated that if n≥2, an operation ∗ between compact convex sets is continuous in the Hausdorff metric, GL(n) covariant, and has the identity property (i.e., K∗{o}=K={o}∗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.
DOI : 10.4171/jems/422
Classification : 52-XX, 39-XX
Keywords: compact convex set, star set, Brunn-Minkowski theory, Minkowski addition, radial addition, Lp​ addition, M-addition, projection, symmetrization, central symmetral, difference body, associativity equation, polynomial volume 
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     author = {Richard J. Gardner and Daniel Hug and Wolfgang Weil},
     title = {Operations between sets in geometry},
     journal = {Journal of the European Mathematical Society},
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     year = {2013},
     doi = {10.4171/jems/422},
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Richard J. Gardner; Daniel Hug; Wolfgang Weil. Operations between sets in geometry. Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2297-2352. doi: 10.4171/jems/422

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