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

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 $\Phi^{-1}$.