Minkowski Geometry

Thompson, Anthony

doi:10.1017/cbo9781107325845

Cited by 283 publications

(240 citation statements)

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“…On the other hand, we may define the altitudes of T in a natural way: For a triangle T with vertices a, b and c, [a, h a ] is called an altitude of T , if, in terms of Birkhoff orthogonality, a − h a is normal to b − c (for Birkhoff orthogonality, cf. [11]). …”

Section: For Everymentioning

confidence: 99%

Ellipsoid characterization theorems

Lángi

2013

Advances in Geometry

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Abstract. 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 int K 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.

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Section: For Everymentioning

confidence: 99%

Ellipsoid characterization theorems

Lángi

2013

Advances in Geometry

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“…Here there are different solutions depending on how (n − 1)-dimensional measure ("area") is defined; see [57].…”

Section: Higher Dimensionsmentioning

confidence: 99%

Antinorms and Radon curves

2006

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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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“…A Minkowski space is a finite dimensional Banach space (M, · ) (see [14]). Thus, up to an isomorphism, every n-dimensional Minkowski space is a normed linear space (R n , · ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Let B H be the Hausdorff metric in C n associated with the metric B induced by the norm · (compare [14]):…”

Section: Preliminariesmentioning

confidence: 99%

Chebyshev sets in hyperspaces over a Minkowski space

Bogdewicz¹,

Dawson²,

Moszyńska

2007

Glas. Mat. Ser. III

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Abstract. In this paper we extend our previous results onČebyšev sets in hyperspaces over a Euclidean n-space to hyperspaces over a Minkowski space.The notion ofČebyšev set has been studied mainly for normed linear spaces (see [4,13]), but it can be considered for arbitrary metric spaces (see [13, Appendix II]). A subset A of a metric space (X, ) is aČebyšev set in this space provided that for every point of X there is a unique nearest point in A. The function ξ A : X → A which assigns to x ∈ X the unique nearest point of A is called metric projection.Cebyšev sets in K n 0 (the space of convex bodies in R n ), K n (the space of nonempty compact convex sets) and O n (the space of compact, strictly convex sets), all endowed with the Hausdorff metric H associated with the Euclidean metric, were studied in [2,5].The present paper is closely related to [2] and [5]. Its purpose is to extend previous results on hyperspaces over a Minkowski space.

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Minkowski Geometry

Cited by 283 publications

References 0 publications

Ellipsoid characterization theorems

Ellipsoid characterization theorems

Antinorms and Radon curves

Chebyshev sets in hyperspaces over a Minkowski space

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