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• W2238687761 abstract "In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincare condition and the measure contraction property. In the second part the notion of uniform smoothness known from the theory of Banach spaces is applied to metric spaces. It is shown that Busemann functions are (quasi-)convex. This implies the existence of a weak soul. In the end further properties are developed to further dissect the soul. In order to understand the influence of curvature on the geometry of a space it helps to develop a synthetic notion. Via comparison geometry sectional curvature bounds can be obtained that demanding that triangles are thinner or fatter. The two classes are called CAT(κ)- and resp. CBB(κ)-spaces. We refer to the book (BH99) and the forthcoming book (AKP) (see also (BGP92, Ots97)). Note that all those notions imply a Riemannian character of the metric space. In particular, the angle between two geodesics starting at a common point is well-defined. Busemann (Bus55, Section 36) study a weaker notion of non-positive curvature which also applies to normed spaces. A similar idea was developed by Pedersen (Ped52) (see also (Bus55, (36.15))) which fits better in the study of Hilbert geometries (KS58). In the recent year a synthetic notion of a lower bound on the Ricci curvature was defined by Lott-Villani (LV09) and Sturm (Stu06). However, their condition include also Finsler manifolds (Oht09, Oht13). The notion of lower curvature bound in the sense of Alexandrov, i.e. CBB(κ)-spaces, is compatible with this Ricci bound (Pet10, GKO13). However, by now there is no known sectional curvature analogue for Finsler-like spaces which is compatible the synthetic Ricci bounds. In this note we present two approaches towards a sectional curvature-type con- dition. The first is the converse of Busemann's non-positive curvature condition. This condition implies a bi-Lipschitz splitting theorem, uniqueness of tangent cones and if the space admits a non-trivial Hausdorff measure then such spaces satisfy doubling and Poincare conditions and even the measure contraction property. This approach rather focuses on the generalized angles formed by two geodesics. The second approach can be seen as a dual to the theory of uniformly convex metric spaces which we call uniform smoothness. This rather weak condition is only powerful in the large. More precisely, we show that Busemann functions associated to rays are (quasi-)convex and that the space has a weak soul. In order to match the theory in the smooth setting we try to develop further assumptions which imply" @default.
• W2238687761 created "2016-06-24" @default.
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• W2238687761 date "2016-01-13" @default.
• W2238687761 modified "2023-09-27" @default.
• W2238687761 title "Sectional curvature-type conditions on Finsler-like metric spaces" @default.
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