Displaying similar documents to “Tiling with smooth and rotund tiles”

Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

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...

An alternative proof of Petty's theorem on equilateral sets

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...

A universal modulus for normed spaces

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.