# On the space of compact diamonds of Lorentzian length spaces

## TL;DR

This paper introduces new product constructions for Lorentzian pre-length spaces and demonstrates that the space of causal diamonds can be endowed with a Lorentzian length space structure, revealing its geodesic and global hyperbolicity properties.

## Contribution

It develops the concepts of taxicab and uniform products for Lorentzian spaces and applies them to analyze the geometry and causal structure of the space of causal diamonds.

## Key findings

- The space of causal diamonds is a Lorentzian length space.
- The space is geodesic for complete X.
- The space is globally hyperbolic for complete X.

## Abstract

In this work we introduce the taxicab and uniform products for Lorentzian pre-length spaces. We further use these concepts to endow the space $D(R\times_T X)$ of causal diamonds with a Lorentzian length space structure, closely relating its causal properties with its geometry as a metric space furnished with its associated Hausdorff distance. Among the general results, we show that this space is geodesic and globally hyperbolic for complete $X$.