Minkowski Endomorphisms

Dorrek, Felix

doi:10.48550/arxiv.1610.08649

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Theorem 22 ([45]1

Dichotomy For the Image Of A Point1

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2016

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“…This turns out to be equivalent to the fact that every function u → f j (K, u) in ( 5) is convex for every 0 ≤ j ≤ n (cf. [18]) which is, in general, not the case. This was first shown in [50] (see also [18]).…”

Section: Theorem 22 ([45]mentioning

confidence: 95%

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Minkowski valuations under volume constraints

Abardia-Evéquoz¹,

Colesanti²,

Gómez³

2016

Preprint

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We provide a description of the space of continuous and translation invariant Minkowski valuations Φ : K n → K n for which there is an upper and a lower bound for the volume of Φ(K) in terms of the volume of the convex body K itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over (n − 1)-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.

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Section: Theorem 22 ([45]mentioning

confidence: 95%

“…(b) For every 1 ≤ k ≤ n, there are exactly a k elements σ ∈ C ′ (n, k) for which (18) holds. (c) I = (1, .…”

Section: Dichotomy For the Image Of A Pointmentioning

confidence: 99%