Hyperconvexity as a generalization of Negative curvature

In this talk, we introduce a novel approach to understanding negative sectional curvature in general metric spaces, drawing inspiration from the concept of the hyperconvex envelope. Hyperconvexity, introduced by Aronszajn and Panitchpakdi in 1956, represents a key concept in metric space theory, characterizing nonexpansive absolute retracts.

An important theory developed by Isbell (1965) and later in another context by Dress (1984) associates to each metric space a hyperconvex envelope via Kuratowski embedding. Dress uses this envelope to define a combinatorial dimension for finite metric spaces.

An intriguing aspect of the hyperconvex hull, as demonstrated by Dress, is its capability to reveal tree-like structures within metric spaces. Specifically, a metric space exhibits tree-like characteristics if its hyperconvex hull exhibits similar properties. Inspired by this characteristic, we explore how hyperconvexity can be interpreted as a generalized form of negative curvature.