Angles in normed spaces

Balestro, Vitor; Horváth, Ákos G.; Martini, Horst; Teixeira, Ralph

doi:10.1007/s00010-016-0445-8

Cited by 36 publications

(19 citation statements)

References 58 publications

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“…Since the quantities we deal with here are affine invariants, we can mostly use the two terms interchangeably. Every o-symmetric convex body K in E d induces a norm on R d , whose closed unit ball centered at o is K, given by known kissing number values correspond to d = 2, 3,4,8,24. For a survey of kissing numbers we refer the interested reader to [9].…”

Section: Introductionmentioning

confidence: 99%

On contact graphs of totally separable domains

2018

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Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this paper, we investigate the Hadwiger and contact numbers of totally separable packings of convex domains, which we refer to as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth convex domain is 4 and the maximum separable contact number of any packing of n translates of a smooth strictly convex domain is 2n − 2 √ n . Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of smooth o-symmetric strictly convex domains by, what we call, Auerbach domains.for every x ∈ R d . We denote the corresponding normed space by (R d , · K ). If K = B d , then we denote the norm · B d simply by · . Thus E d = (R d , · ). A d-dimensional convex body K is said to be smooth if at every point on the boundary bd K of K, the body K is supported by a unique hyperplane of E d and strictly convex if the boundary of K contains no nontrivial line segment. Given d-dimensional convex bodies K and L, their Minkowski sum (vector sum) is denoted by K + L and defined aswhich is a convex body in E d .The kissing number problem asks for the maximum number k(d) of non-overlapping translates of B d that can touch B d in E d . Clearly, k(2) = 6. However, the value of k(3) remained a mystery for several hundred years and caused a famous argument between Newton and Gregory in the 17th century. To date, the only *

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Section: Introductionmentioning

confidence: 99%

On contact graphs of totally separable domains

2018

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“…This type of angular measure is very useful in studying packings of unit balls [ 2 , 8 ]. Angular measures with other properties have been proposed; see the survey [ 1 , Section 4] for an overview. An angular measure is called a B-measure [ 3 ] if for every closed arc C of that contains no opposite points of , and whose endpoints x and y satisfy .…”

Section: Introductionmentioning

confidence: 99%

Angular measures and Birkhoff orthogonality in Minkowski planes

2020

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Let x and y be two unit vectors in a normed plane R 2 . We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankhänel 2011) is an angular measure µ on the unit circle for which µ(C) = π/2 whenever C is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a Bmeasure. * Part of the research was carried out while MN was a member of János Pach's chair of DCG at EPFL, Lausanne

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“…The fundamental difference between non-Euclidean Minkowski spaces and the Euclidean space is the absence of an inner product, and thus the notions of angle and orthogonality do not exist in the usual sense. Nevertheless, several types of orthogonality can be defined (see [1], [2], and [6] for an overview; for angles we refer to [9]), with isosceles and Birkhoff orthogonalities being the most prominent examples. We say that y is Birkhoff orthogonal to x, denoted x ⊥ B y, when x ≤ x + αy for any α ∈ R.…”

Section: Introductionmentioning

confidence: 99%

Geometry of Simplices in Minkowski Spaces

Leopold

Martini

2018

Results Math

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There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We present new generalizations of well known properties of Euclidean simplices. These results refer to analogues of circumcenters, Euler lines, and Feuerbach spheres of simplices in normed spaces. Using duality, we also get natural theorems on angular bisectors as well as inand exspheres of (dual) simplices.

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Angles in normed spaces

Cited by 36 publications

References 58 publications

On contact graphs of totally separable domains

On contact graphs of totally separable domains

Angular measures and Birkhoff orthogonality in Minkowski planes

Geometry of Simplices in Minkowski Spaces

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