# Geometry of Minkowski Planes and Spaces -- Selected Topics

### Hinweis

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http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200900226

### Kurzfassung in Englisch

The results presented in this dissertation refer to the geometry of Minkowski
spaces, i.e., of real finite-dimensional Banach spaces.

First we study geometric properties of radial projections of
bisectors in Minkowski spaces, especially the relation between the
geometric structure of radial projections and Birkhoff
orthogonality. As an application of our results it is shown that for
any Minkowski space there exists a number, which plays somehow the
role that $\sqrt2$ plays in Euclidean space. This number is referred
to as the critical number of any Minkowski space. Lower and upper
bounds on the critical number are given, and the cases when these
bounds are attained are characterized. Moreover, with the help of
the properties of bisectors we show that a linear map from a normed
linear space $X$ to another normed linear space $Y$ preserves
isosceles orthogonality if and only if it is a scalar multiple of a
linear isometry.

Further on, we examine the two tangent segments from any exterior
point to the unit circle, the relation between the length of a chord
of the unit circle and the length of the arc corresponding to it,
the distances from the normalization of the sum of two unit vectors
to those two vectors, and the extension of the notions of
orthocentric systems and orthocenters in Euclidean plane into
Minkowski spaces. Also we prove theorems referring to chords of
Minkowski circles and balls which are either concurrent or parallel.
All these discussions yield many interesting characterizations of
the Euclidean spaces among all (strictly convex) Minkowski spaces.

In the final chapter we investigate the relation between the length
of a closed curve and the length of its midpoint curve as well as
the length of its image under the so-called halving pair
transformation. We show that the image curve under the halving pair
transformation is convex provided the original curve is convex.
Moreover, we obtain several inequalities to show the relation
between the halving distance and other quantities well known in
convex geometry. It is known that the lower bound for the geometric
dilation of rectifiable simple closed curves in the Euclidean plane
is $\pi/2$, which can be attained only by circles. We extend this
result to Minkowski planes by proving that the lower bound for the
geometric dilation of rectifiable simple closed curves in a
Minkowski plane $X$ is analogously a quarter of the circumference of
the unit circle $S_X$ of $X$, but can also be attained by curves
that are not Minkowskian circles. In addition we show that the lower
bound is attained only by Minkowskian circles if the respective norm
is strictly convex. Also we give a sufficient condition for the
geometric dilation of a closed convex curve to be larger than a
quarter of the perimeter of the unit circle.

 Schlagwörter Birkhoff orthogonality Schlagwörter James orthogonality Schlagwörter Minkowski Geometry Schlagwörter Minkowski plane Schlagwörter Minkowski space Schlagwörter characterizations of Euclidean planes Schlagwörter characterizations of inner product spaces Schlagwörter convex geometry SWD Schlagworte Konvexität SWD Schlagworte Mathematik DDC Klassifikation 510 Institution(en) Hochschule TU Chemnitz Fakultät Fakultät für Mathematik Betreuer Prof. Dr. Horst Martini Gutachter Prof. Dr. Horst MartiniProf. Dr. Gunter WeißProf. Dr. Eike Hertel Dokumententyp Dissertation Sprache Englisch Tag d. Einreichung (bei der Fakultät) 01.10.2008 Tag d. Verteidigung / Kolloquiums / Prüfung 13.11.2008 Veröffentlichungsdatum (online) 03.02.2009 persistente URN urn:nbn:de:bsz:ch1-200900226

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