We say that a convex body R of a d-dimensional real normed linear space M d is reduced, if ∆(P ) < ∆(R) for every convex body P ⊂ R different from R. The symbol ∆(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M d . We establish a number of properties of reduced bodies in M 2 . They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M 2 is larger than ∆(R) for all directions strictly between two fixed directions and equals ∆(R) for these two directions.
We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are 3 2 . We also prove that C permits to inscribe a triangle whose sides are of lengths at least 3 2 in the positive orientation, or that they are of lengths at least 3 2 in the negative orientation. The ratio 3 2 in both the theorems is best possible.
Mathematics Subject Classification (2000). 52A10, 52A21.We start by recalling the notion of the Minkowski plane. Let C be a convex body in Euclidean plane E 2 and let z ∈ int(C). We define the oriented distance δ C,z (a, b) of points a and b as δ C,z (a, b) = |ab|/|zw|, where w ∈ bd(C)such that the vectors −→ zw and −→ ab have the same orientation (the symbol | | stands for the Euclidean distance). We call C the unit disk and z the origin. The plane with the oriented distance is called Minkowski plane. It is a special case of Minkowski space. Many properties of Minkowski plane are presented in [5,6,9]. When C is centrally symmetric with z as its center, we get the two-dimensional normed plane.The well-known problem if the self-circumference of an arbitrary Minkowski unit disk is at least 6 (see [2,5,7]) caused many questions about "large" inscribed polygons in the unit disk. Our particular interest is in "large" inscribed and also contained triangles. We continue our research from [4].If we go on the boundary of a triangle according to the positive (respectively: negative) orientation and if the oriented lengths of its sides are equal, then we call the triangle equilateral in the positive (respectively: negative) orientation. We also simply say orientation when it is positive. Since the Minkowski functional is non-symmetric in general, equilateral triangles usually are not
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