Summary
LetK d denote the cone of all convex bodies in the Euclidean spaceK d. The mappingK ↦ h K of each bodyK ∈ K d onto its support function induces a metricδ w onK d byδ" w (K, L)≔∥h L −h K ∥ w where ∥⋅∥ w is the Sobolev I-norm on the unit sphere\(\mathbb{S}^{d - 1} \subset \mathbb{E}^d \). We callδ w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.
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