By David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin Shen

Finsler geometry generalizes Riemannian geometry in precisely an analogous approach that Banach areas generalize Hilbert areas. This publication provides expository bills of six very important subject matters in Finsler geometry at a degree compatible for a different issues graduate direction in differential geometry. The members think about matters relating to quantity, geodesics, curvature and mathematical biology, and contain a number of instructive examples.

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Let X be a two-dimensional normed space with unit disc B and area form Ω ∈ Λ2 X ∗ . Of all convex bodies in X with a given perimeter the one that encloses the largest area is, up to translations, a dilate of I := (iΩ B)∗ . VOLUMES ON NORMED AND FINSLER SPACES 33 Proof. Let K ⊂ X be a convex body and let B (∂K) = 2V (K, I) be its perimeter. By Minkowski’s mixed volume inequality, we have B (∂K) 4 2 = V (K, I)2 ≥ λ(K)λ(I) with equality if an only if K and I are homothetic. Thus, the area enclosed by K is maximal for a given perimeter if and only if K is a dilate of I.

VOLUMES ON NORMED AND FINSLER SPACES 19 the k-dimensional normed space “spanned by a”, a := {x ∈ X : a ∧ x = 0} ⊂ X, (provided with the induced norm), and compute µ a (a). Thus, once we have chosen a way to define volume in all finite-dimensional normed spaces, we have a way to associate to each norm on an n-dimensional vector space X a family of k-volume densities, with 1 ≤ k ≤ n. The Busemann k-volume densities. Let X be a normed space of dimension n with unit ball B, and let k be a positive integer less than n.

12 characterizes the Holmes–Thompson definition. However, the following question remains open. Problem [Thompson 1996]. Is the Holmes–Thompson definition of volume characterized by the fact that the area of the unit sphere of a normed space equals the area of the unit sphere of its dual ? The first result extending Gol¸ab’s theorem to higher dimension is the following sharp upper bound for the Busemann area of a unit sphere. 13 [Busemann and Petty 1956]. The Busemann area of the unit sphere of an n-dimensional normed space is at most 2nεn−1 with equality if and only if B is a parallelotope.