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

It is surprising that there are almost no results on the precise location of (all) minimal enclosing balls, circumballs, and circumcenters of simplices in finite-dimensional real Banach spaces. In this paper and a subsequent second part of it we give the starting point in this direction, also for computational investigations. More precisely, we present the first thorough study of these topics for triangles in arbitrary normed planes. In the present Part I we lay special emphasize on a complete description of possible locations of the circumcenters, and as a needed tool we give also a modernized classification of all possible shapes of the intersection that two homothetic norm circles can create. Based on this, we give in Part II the complete solution of the strongly related subject to find all minimal enclosing discs of triangles in arbitrary normed planes. (C) 2012 Elsevier B.V. All rights reserved.