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

AbstractIt 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

Geometric Algorithms for Minimal Enclosing Disks in Strictly Convex Normed Planes

With the geometric background provided by Alonso, Martini, and Spirovaon the location of circumcenters of triangles in normed planes, we show the validity of the Elzinga--Hearn algorithm and the Shamos--Hoey algorithm for solving the minimal enclosing disk problem in strictly convex normed planes

Kolmiot konveksin etäisyyden tasoissa

The article deals with a plane equipped with a convex distance function. We extend the notions of equilateral and acute triangles and consider circumcenters of such triangles.Peer reviewe

Huomioita keskuspisteistä normitasoissa

Peer reviewe

Ball hulls, ball intersections, and 2-center problems for gauges

The notions of ball hull and ball intersection of nite sets, important in Banach space theory, are extended from normed planes to generalized normed planes, i.e., to (asymmetric) convex distance functions which are also called gauges. In this more general setting we derive various new results about these notions and their relations to each other. Further on, we extend the known 2-center problem and a modified version of it from the Euclidean situation to norms and gauges or, in other words, from Euclidean circles to arbitrary closed convex curves. We derive algorithmical results on the construction of ball hulls and ball intersections, and computational approaches to the 2-center problem with constrained circles and, in case of strictly convex norms and gauges, for the fixed 2-center problem are also given

New Moduli for Banach Spaces

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti of right-angled triangles (defined via so-called quasi-orthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander type results, involving generalizations of the modulus of convexity and the modulus of Bana\'{s}