Visibility in semi-convex spaces

Abstract

We introduce the notion of a semi-convex space as a unifying framework for the treatment of various notions of convexity in the plane. Semi-convex spaces are a generalization of convexity spaces that are more appropriate for investigating issues of visibility. We define the notion of visibility within the general framework of semi-convex spaces, and investigate the relationship between visibility, kernels, and skulls. We prove the Kernel Theorem and the Cover Kernel Theorem, both of which relate kernels and skulls. Based on these results for semi-convex spaces we prove a theorem about metrics in the plane and demonstrate the utility of our theory with two examples of semi-convex spaces based on geodesic convexity and staircase convexity.