Minimal enclosing discs, circumcircles, and circumcenters in normed planes (Part I)

Geometry of Simplices in Minkowski Spaces

“…For two-dimensional simplices, we are able to extract additional information. Combining Theorem 2.1 and Theorem 2.2, we obtain that the circumcenter of any 2-simplex ABC in a Minkowski plane can only lie in the interior or boundary of the shaded region in Figure 2 which is defined by the lines through edge midpoints, as established by other (longer) arguments in [3,Theorem 4.1.]. In fact, in [3] the stronger statement is proved that for any point M in Figure 2 a norm exists which makes M a circumcenter of the 2-simplex ABC if and only if M lies in the shaded region.…”

“…Combining Theorem 2.1 and Theorem 2.2, we obtain that the circumcenter of any 2-simplex ABC in a Minkowski plane can only lie in the interior or boundary of the shaded region in Figure 2 which is defined by the lines through edge midpoints, as established by other (longer) arguments in [3,Theorem 4.1.]. In fact, in [3] the stronger statement is proved that for any point M in Figure 2 a norm exists which makes M a circumcenter of the 2-simplex ABC if and only if M lies in the shaded region. It is not hard to see why the converse in this statement is true; the convex hull of triangle ABC and its image A B C with respect to a half-turn around a point M located in the shaded region is a suitable unit ball (the antipodal pairs A, A , B, B , and C, C are contained in its boundary).…”

“…Inspired by the work [3] of Alonso, Martini and Spirova for the planar case (see also [28]), we establish results about the location of circumcenters in the d-dimensional case, i.e., about the centers of circumspheres as defined above. Incidentally, this also proves some statements about the planar case in a more condensed form than in [3]. We define the medial hyperplane between a facet F of a d-dimensional simplex and the opposite vertex A as the hyperplane bisecting every segment from A to a point in F .…”

“…Before presenting our results on circumcenters of simplices, we underline that we mean the centers of circumspheres (or -balls) of simplices, i.e., of Minkowskian spheres containing all the vertices of the respective simplex (see, e.g., [3]). A related, but different notion is that of minimal enclosing spheres of simplices, sometimes also called circumspheres (cf., e.g., [4]); this notion is not discussed here.…”

“…Circumcenters and Chebyshev sets are of interest in adjacent disciplines such as location science and approximation theory. Further on, solutions to questions from Elementary Geometry in normed spaces very often yield a tool and form the first step for attacking problems in the spirit of Discrete and Computational Geometry in such spaces (see, e.g., [3,4]). And of course it is an interesting task for geometers to generalize notions like orthocentricity (cf.…”