Reshetnyak Majorisation and discrete upper curvature bounds for Lorentzian length spaces

Abstract

We present an analogue to the Majorisation Theorem of Reshetnyak in the setting of Lorentzian length spaces with upper curvature bounds: given two future-directed timelike rectifiable curves $\alpha$ and $\beta$ with the same endpoints in a Lorentzian length space $X$, there exists a convex region in $\mathbb{L}^2(K)$ bounded by two future-directed causal curves $\bar \alpha$ and $\bar \beta$ with the same endpoints and a 1-anti-Lipschitz map from that region into $X$ such that $\bar \alpha$ and $\bar \beta$ are respectively mapped $\tau$-length-preservingly onto $\alpha$ and $\beta$. A special case of this theorem leads to an interesting characterisation of upper curvature bounds via four-point configurations which is truly suitable for a discrete setting.