Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

TL;DR

This paper studies the continuity properties of a mapping from hv-convex planar sets to their associated generalized conic functions, revealing how convergence in sets relates to convergence of these functions and implications for geometric tomography.

Contribution

It establishes the conditions under which the conic functions are continuous with respect to Hausdorff convergence and characterizes the sets determined by their coordinate X-rays.

Findings

Hausdorff convergence implies convergence of conic functions in supremum and L1 norms

The inverse mapping is upper semi-continuous, enabling approximation of sets from their conic functions

Convex bodies determined by their coordinate X-rays are characterized by lower semi-continuity of the inverse mapping

Abstract

In the paper we investigate the continuity properties of the mapping $\Phi$ which sends any non-empty compact connected hv-convex planar set $K$ to the associated generalized conic function $f_K$. The function $f_K$ measures the average taxicab distance of the points in the plane from the focal set $K$ by integration. The main area of the applications is the geometric tomography because $f_K$ involves the coordinate X-rays' information as second order partial derivatives \cite{NV3}. We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the $L_1$-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that $\Phi^{-1}$ is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if $f_L$ is close to $f_K$ then $L$ must be close to an element $K'$ such that $f_{K}=f_{K'}$. Therefore $K$ and $K'$ have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies $K$ for which $f_K$ is a point of lower semi-continuity for $\Phi^{-1}$.