Problems Related To The Circle In Minkowski Planes

In 2009, H. Martini and Senlin Wu proved that a Minkowski plane (i.e., a real two-dimensional normed linear space) X is a Euclidean plane (a real two-dimensional inner product space) if and only if the two piece arcs (two chords, resp.) between any two parallel chords intersecting the unit circle have the same length. In this paper we attempt to ask following questions: For an arbitrary Minkowski plane, and two arbitrary concurrent chords of the unit circle which are of equal lengths, whether there is one pair of opposite circular arcs (chords, resp.) bounded by these two chords are of equal lengths.First, this paper reviews Minkowski geometry (i.e., geometry of real finite-dimensional Banach spaces) and its development, surveys the main existing results, and presents the required preliminaries, including definitions and results concerning triangle inequality in Minkowski spaces, strict convexity, and monotonicity lemma.From the "elementary" geometry view point, the relation between a chord of the unit circle and the circular arc corresponding to it is discussed. We gave a new proof of the fact that a Minkowski plane is the Euclidean plane if and only if any two chords of the unit circle which are of equal lengths are corresponding to a pair of circular arcs of equal length. Based on this result, it is proved that a Minkowski plane is Euclidean if and only if two concurrent chords of equal lengths divide the unit circle of the underlying plane into two pairs of opposite arcs (chords, resp.), one of which are of equal lengths.