Rotation equivariant Minkowski valuations

By Rolf Schneider, Franz E. Schuster. 2012

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The projection body operator \Pi, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that \Pi\ maps the set of polytopes in Rn into itself. We show that \Pi\ is the only non-trivial operator with these properties.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/27b32f4fe57e20cb18373b9bd82d98049bb70c9e

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Valuations on Sobolev spaces (2012, background): “See [25, 30, 47, 48] for information on the classical theory of valuations on convex bodies and [1, 2, 3, 4, 5, 9, 16, 21, 20, 22, 23, 33, 34, 35, 38, 39, 51, 53, 54, 58] for some of the more recent results.”
Projection bodies in complex vector spaces (2011, background): “See [6, 18, 24, 25, 26, 27, 28, 44, 46, 47, 48] for more information on Minkowski valuations.”
GL(n) contravariant Minkowski valuations (2012, result): “First results on Minkowski valuations which are rigid motion equivariant were obtained by Schneider [32] in the 1970s; see [14, 34, 37, 38] for recent extensions of these results.”
Valuations and Busemann{Petty Type Problems (2008, background): “Previous investigations show that these new properties lead to a large class of valuations, see [28,57,59,62].”
Minkowski valuations intertwining the special linear group (2012, unknown): “, [24–26,29,30,35,37–39,56–58]).”
Minkowski valuations and generalized valuations (2015, background): “[57]), which we state here under the additional assumption of SO(n) equivariance.”
Fisher information and matrix-valued valuations (2011, result): “See [24, 27] for information on the classical theory of valuations on convex sets and [1–5,9,11,16,19–22,28– 31, 33, 34, 43–45] for some of the more recent results.”
A classification of $\operatorname{SL}(n)$ invariant valuations (2010, background): “Since then investigations of valuations have been an active and prominent part of mathematics (see [1] – [8], [16], [19], [20], [24], [25], [31] – [34], [53] for some of the more recent results).”
Difference bodies in complex vector spaces (2012, unknown): “For the minimal index k0 and the maximal index k1 with fk 6= 0, it was proved in [41] that fk0 and fk1 are support functions.”
Minkowski Areas and Valuations (2010, background): “of elementary mixed volumes, valuations have become a critical notion (see [27, 31, 47] and see [1, 2, 3, 4, 5, 6, 11, 15, 29, 30, 49, 51, 52, 53] for some of the more recent contributions).”
Crofton Measures and Minkowski Valuations (2010, background): “First results on a special class of Minkowski valuations were obtained by Schneider [42] in the 1970s, but only through the recent seminal work of Ludwig [29, 31] classifications of convex and star body valued valuations have become the focus of increased attention, see [21, 22, 24, 32, 44, 47].”
Star body valued valuations (2009, background): “Ludwig’s seminal work [21], [23]–[25] showed that fundamental operators in convexity and geometric tomography can be classified as convex or star body valued valuations which are compatible with the general linear group (see also [13], [42]).”
$\mathrm{GL}(n)$ contravariant Minkowski valuations (2012, background): “First results on Minkowski valuations which are rigid motion equivariant were obtained by Schneider [32] in the 1970s (see [14, 34, 37, 38] for recent extensions of these results).”