OPERATIONS BETWEEN SETS IN GEOMETRY

By Richard J. Gardner, Daniel Hug, Wolfgang H. Weil. 2012

109 citations recorded by the source corpus.

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 R n . It is proved that if n 2, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdor metric, GL(n) covariant, and associative if and only if it is Lp addition for some 1 p 1 . It is also demonstrated that if n 2, an operation between compact convex sets is continuous in the Hausdor metric, GL(n) covariant, and has the identity property (i.e., Kf og = K =fog 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 eect. Several other new lines of investigation are followed. A relatively little-known but semi- nal operation called M-addition is generalized and systematically studied for the rst time. Geometric-analytic formulas that characterize continuous andGL(n)-covariant operations be- tween compact convex sets in terms of M-addition are established. It is shown that if n 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 Hausdor metric, GL(n) co- variant, and translation invariant if and only if it is of the form DK for some 0, where DK = K +( K) is the dierence body of K. The term \polynomial volume" is introduced for the property of operations between compact convex or star sets that the volume of rK sL, r;s 0, is a polynomial in the variables r and s. It is proved that if n 2, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdor metric, GL(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/9fdf14799ed5eb3772d31cc10b991782396ca9ba

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Citations in the corpus

The dual Orlicz–Brunn–Minkowski theory ☆ (2014, methodology): “We refer the reader to the introductions in [15, 16] and to [37, Chapter 9] for more information and references.”
The dual Minkowski problem for negative indices (2017, unknown): “Although the duality demonstrated in the dual Brunn-Minkowski theory is only heuristic, it has provided numerous strikingly similar (formally), yet significant concepts and results, see, for example, Gardner [15], Gardner, Hug & Weil [16], Haberl [18], Haberl & Ludwig [20], and Zhang [46].”
Existence of solutions to the even dual Minkowski problem (2018, background): “Quoting from Gardner, Hug & Weil [19]:“The dual Brunn-Minkowski theory can count among its successes the solution of the Busemann-Petty problem in [16], [20], [41], and [68].”
The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems (2017, methodology): “More properties and historical remarks for the M -addition, such as convexity, GL(n) covariance, homogeneity and monotonicity, can be founded in [15, 16, 35, 36].”
Symmetrization in geometry (2016, methodology): “These are examples of 0-symmetrizations not covered by the results in [9] or the present paper.”
Necessary subspace concentration conditions for the even dual Minkowski problem (2017, background): “ and references. There are two far-reaching extensions of the classical Brunn-Minkowski theory, both arising basically by replacing the classical Minkowski-addition by another additive operation (cf. [22, 23]). The ﬁrst one is the Lp addition introduced by Firey (see, e.g., [18]) which leads to the ”
Orlicz–John ellipsoids (2014, background): “Very recently, one essential obstacle in the development of Orlicz Brunn–Minkowski theory, what is the lack of a notion corresponding to Lp addition, has been smoothed by Gardner, Hug and Weil [14,15].”
Mixed integrals and related inequalities (2012, background): “In fact, in [6] the authors consider the problem of characterizing the Minkowski addition.”
THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES (2013, background): “ of ∗(K,L). In the following list, it is assumed that C and D are suitable classes for the property under consideration. The properties are supposed tohold forall appropriate K,L,N,Kj,Lj,Kij ∈ C. See [10] for further properties and information. 1. (Commutativity (m= 2 only)) K∗L= L∗K. 2. (Associativ”