Blaschke- and Minkowski-endomorphisms of convex bodies

By Markus Kiderlen. 2006

70 citations recorded by the source corpus.

We consider maps of the family of convex bodies in Euclidean d-dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms. The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/aef6e2650aff4c688a0449944da558cfc3b3d2ce

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

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).”
Rotation equivariant Minkowski valuations (2012, background): “This class of operators was introduced and investigated by the first author, see [24, 25], and more recently studied by Kiderlen [ 9 ].”
Valuations on Sobolev spaces (2012, result): “See [12, 18, 26, 29, 41, 42, 43, 52] for some of the recent results involving Blaschke addition and, in particular, Haberl [21], where a classification of Blaschke valuations on convex bodies was obtained.”
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].”
Volume inequalities and additive maps of convex bodies (2006, result): “Among other results, he obtained a complete classification of all such maps in R2. Kiderlen [ 15 ] continued these investigations, and extended Schneider’s 218”
$\mathrm{GL}(n)$ 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).”
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].”
Convolutions and multiplier transformations of convex bodies (2007, methodology): “The following conjecture appears implicitly in [27] and [ 13 ].”
General $L_p$ affine isoperimetric inequalities (2008, background): “First results on convex body valued valuations were obtained by Schneider [41] in the 1970s, where the addition of convex bodies in (1) is Minkowski addition dened by h(K +L; ) = h(K; ) +h(L; ), see also [ 17 , 43, 45].”
Difference bodies in complex vector spaces (2012, background): “Some results on Minkowksi valuations not described in this paper can be found, for instance, in [18, 19, 21, 32, 33, 42, 43, 45].”
Minkowski valuations and generalized valuations (2015, methodology): “Using this description of valuations in Val1, Kiderlen [36] proved that if Φ1 ∈ MVal1 is SO(n) equivariant, then there exists a uniquely determined zonal ν1 ∈ C o (S) of order at most 2 such that hΦ1K = hK ∗ ν1 (3.”