Inner and Outer j-Radii of Convex Bodies in Finite-Dimensional Normed Spaces.
V. Klee, P. Gritzmann (1992)
Discrete & computational geometry
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V. Klee, P. Gritzmann (1992)
Discrete & computational geometry
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Tomasz Kobos (2013)
Annales Polonici Mathematici
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The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth...
Milman, Vitali (1998)
Documenta Mathematica
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Yunbai Dong, Qingjin Cheng (2013)
Studia Mathematica
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Let 𝓐 be a compatible collection of bounded subsets in a normed linear space. We give a characterization of the following generalized Mazur intersection property: every closed convex set A ∈ 𝓐 is an intersection of balls.
T. Figiel (1976)
Compositio Mathematica
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Carlos Benítez, Krzysztof Przesławski, David Yost (1998)
Studia Mathematica
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We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.
Victor Klee, Libor Veselý, Clemente Zanco (1996)
Studia Mathematica
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For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation...
MANUEL FERNÁNDEZ and MARÍA L. SORIANO CARLOS BENÍTEZ (2000)
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales
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Carlos Benítez Rodríguez (1989)
Revista Matemática de la Universidad Complutense de Madrid
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Orthogonality in inner products is a binary relation that can be expressed in many ways without explicit mention to the inner product of the space. Great part of such definitions have also sense in normed linear spaces. This simple observation is at the base of many concepts of orthogonality in these more general structures. Various authors introduced such concepts over the last fifty years, although the origins of some of the most interesting results that can be obtained for these generalized...
Pap, Endre (1996)
Novi Sad Journal of Mathematics
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Lin, C.-S. (1992)
International Journal of Mathematics and Mathematical Sciences
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Stanisław Kryński (1993)
Studia Mathematica
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Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.
Günter Ewald, Geoffrey C. Shephard (1966)
Mathematische Zeitschrift
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C.-S. Lin (2005)
Colloquium Mathematicae
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We first introduce a notion of (a,b,c,d)-orthogonality in a normed linear space, which is a natural generalization of the classical isosceles and Pythagorean orthogonalities, and well known α- and (α,β)-orthogonalities. Then we characterize inner product spaces in several ways, among others, in terms of one orthogonality implying another orthogonality.