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

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In the paper we investigate the continuity properties of the mapping Φ which sends any non-empty compact connected hv-convex planar set K to the associated generalized conic function fₖ. The function fₖ 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 fₖ involves the coordinate X-rays' information as second order partial derivatives NV3. We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the L₁-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 Φ⁻¹ is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if fₗ is close to fₖ then L must be close to an element K' such that fₖ=fₖ'. 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 fₖ is a point of lower semi-continuity for Φ⁻¹.