Symmetric orthogonality and non-expansive projections in metric spaces

TL;DR

This paper extends the concepts of symmetric orthogonality and non-expansive projections from linear spaces to metric spaces, showing their equivalence in Busemann non-positively curved spaces and characterizing $CAT(0)$-spaces under certain conditions.

Contribution

It generalizes known linear space results to metric spaces and establishes conditions under which Busemann non-positive curvature implies $CAT(0)$-space structure.

Findings

Symmetric orthogonality and non-expansive projections are equivalent in Busemann non-positively curved spaces.

Every Busemann non-positively curved space with uniquely geodesic tangent cones and non-expansive projections is a $CAT(0)$-space.

Abstract

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it is shown that every space having non-positive curvature in the sense of Busemann is a $CAT(0)$-space provided that its tangent cones are uniquely geodesic and their nearest point projections onto convex are non-expansive.