Conics in normed planes

We study the generalized analogues of conics for normed planes by using the following natural approach: It is well known that there are different metrical definitions of conics in the Euclidean plane. We investigate how these definitions extend to normed planes, and we show that in this more general framework these different definitions yield, in almost all cases, different classes of curves.Comment: 12 pages, 8 figure

Orthocentric Systems in Minkowski Planes

A new way to define the notion of \C-orthocenter will be displayed by studying some propierties of four points in the plane which allows to extend the notion of Euler's line, the Six Point Circles and the three-circles theorem, for normed planes. In the present paper (which can be regarded as extention of \cite{M-WU} in Minkowski plane in general) we derive several characterizations of the Euclidianity of the plane.Comment: 14 pages, 7 figure

Ellipsoid characterization theorems

In this note we prove two ellipsoid characterization theorems. The first one is that if K is a convex body in a normed space with unit ball M, and for any point p ∉ K and in any 2-dimensional plane P intersecting intK and containing p, there are two tangent segments of the same normed length from p to K, then K and M are homothetic ellipsoids. Furthermore, we show that if M is the unit ball of a strictly convex, smooth norm, and in this norm billiard angular bisectors coincide with Busemann angular bisectors or Glogovskij angular bisectors, then M is an ellipse