Geodesic
Inner Products for Convex Bodies
Journal of convex analysis, Tome 28 (2021) no. 4, pp. 1249-1264
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We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hörmander and Radström. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
Classification : 52A20, 52A27, 05C05
Mots-clés : Inner product, convex body, Minkowski linear functionals, ecological niche 
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     author = {D. Bryant and P. Cioica-Licht and L. O. Clark and R. Young},
     title = {Inner {Products} for {Convex} {Bodies}},
     journal = {Journal of convex analysis},
     pages = {1249--1264},
     year = {2021},
     volume = {28},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2021_28_4_JCA_2021_28_4_a14/}
}
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D. Bryant; P. Cioica-Licht; L. O. Clark; R. Young. Inner Products for Convex Bodies. Journal of convex analysis, Tome 28 (2021) no. 4, pp. 1249-1264. http://geodesic.mathdoc.fr/item/JCA_2021_28_4_JCA_2021_28_4_a14/