We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.

We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations. Mathematics Subject Classification (2000). 46B20, 46C15, 52A21.

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In this paper we consider two notions that have been discovered and rediscovered by geometers and analysts since 1917 up to the present day: Radon curves and antinorms. A Radon curve is a special kind of centrally symmetric closed convex curve in the plane. A Radon plane is a normed plane obtained by using a Radon curve as (the boundary of) the unit ball. Many known results in Euclidean geometry also hold for Radon planes, for example the triangle and parallelogram area formulas, certain theorems on angular bisectors, the area formula of a polygon circumscribed about a circle, certain isoperimetric inequalities, and the non-expansive property of certain non-linear projections. These results may be further generalized to arbitrary normed planes if we formally change the statement of the result by referring in some places to the antinorm instead of the norm. The antinorm is a norm dual to the norm of an arbitrarily given normed plane, although it lives in the same plane as the original norm. It is the purpose of this mainly expository paper to give a list of results on antinorms that generalize results true for Radon norms, and in many cases characterize Radon norms among all norms in the plane. Many of the results are old, well-known, and have often been rediscovered. However, for most of the results we give streamlined proofs. Also, some of the characterizations of Radon curves given here seem not to have appeared previously in print.Comment: 24 pages, 7 figure

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This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg-Markus-Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalization of this problem for the case of unbounded convex bodies; visibility and inner illumination of convex bodies; primitive illuminating systems for convex bodies; illumination and visibility of families of convex bodies; clouds formed by translates or homothetic copies of a convex body; miscellaneous results.Mathematics Subject Classification (1991). 52A37, 52A40.