Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

Lassak, Marek; Martini, Horst

doi:10.1007/s00025-014-0384-4

Cited by 19 publications

(16 citation statements)

References 34 publications

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“…The first statement directly follows from the fact that K is of constant width with respect to C iff K − K = 1/2D(K, C)(C − C), and (ii) as well as (iii) directly follow from the fact that w(K, C) = 2w(K, C − C) and D(K, C) = 2D(K, C − C). The fourth statement is a direct corollary out of (i) and (ii), while the others follow from (ii) and (iii), taking into account that all those statements are well known for the case that C = −C (see [10], [19], and [21]).…”

Section: Completeness and Reducednessmentioning

confidence: 84%

“…The proposition below is taken from [19,Corollary 7] and shows a quite similar structure for the reducedness of simplices as the one given in Proposition 3.6 for completeness.…”

Section: Completeness and Reducednessmentioning

confidence: 99%

“…[1] and [18]). As first shown in [17], the notion of reducedness can be carried over to Minkowski spaces, too; existing results and interesting research problems are presented in [19]. Since, in general, in such spaces complete bodies need not be reduced (see [22]) and one can construct reduced bodies which are not complete, the question is how these two classes are related to each other.…”

Section: Introductionmentioning

confidence: 99%

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Is a complete, reduced set necessarily of constant width?

Brandenberg

Merino

Jähn

et al. 2019

Advances in Geometry

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Is it true that a convex body K being complete and reduced with respect to some gauge body C is necessarily of constant width, i. e., satisfies K − K = ρ(C − C) for some ρ > 0? We prove this implication for several cases including the following: if K is a simplex and or if K possesses a smooth extreme point, then the implication holds. Moreover, we derive several new results on perfect norms.2010 Mathematics Subject Classification. 46B20, 52A20, 52A21, 52A40, 52B11.

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Section: Completeness and Reducednessmentioning

confidence: 84%

“…The proposition below is taken from [19,Corollary 7] and shows a quite similar structure for the reducedness of simplices as the one given in Proposition 3.6 for completeness.…”

Section: Completeness and Reducednessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Is a complete, reduced set necessarily of constant width?

Brandenberg

Merino

Jähn

et al. 2019

Advances in Geometry

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“…In general, the norm induced by the isoperimetrix of a unit ball (itself a convex body symmetric to the origin) is called antinorm, see [24]. A convex body is called reduced with respect to a given norm of the ambient space if any properly contained convex body has smaller minimum width (for reduced bodies in Minkowski spaces we refer to the survey [15]). M 2 (B), whereB is the isoperimetrix of B, i.e., the dual rotated by an angle of π/2 about the origin.…”

Section: Dualitymentioning

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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“…In any Minkowski space X, bodies of constant width are precisely determined by the condition ∆ (K) = δ (K), and a convex body is called (diametrically) complete in X if it is not properly contained in a compact set of the same diameter. In a sense dually, a convex body R is said to be reduced in X if ∆ (C) < ∆ (R) holds for any convex body C properly contained in R. There are interesting open problems about complete and reduced bodies in Minkowski spaces (see the survey [2]), and even for the Euclidean norm surprisingly elementary questions are still unsolved (cf. [1]).…”

mentioning

confidence: 99%

Complete sets need not be reduced in Minkowski spaces

Martini

2015

Beitr Algebra Geom

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It is well known that in n-dimensional Euclidean space (n ≥ 2) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For n-dimensional Minkowski spaces, this coincidence is no longer true if n ≥ 3. Thus, the question occurs whether for n ≥ 3 any complete set is reduced. Answering this in the negative for n ≥ 3, we construct (2 k − 1)-dimensional (k ≥ 2) complete sets which are not reduced.Keywords bodies of constant width · (diametrically) complete sets · Hadamard matrix · Minkowski Geometry · reduced body · Walsh matrix Mathematics Subject Classification (2000) 46B20 · 52A20 · 52A21 · 52A40 · 52B11 IntroductionWe denote by X = (R n , · ) an n-dimensional normed or Minkowski space, i.e., an n-dimensional real Banach space with origin o. We will use the common abbreviations aff and conv for affine and convex hull, respectively. A convex body in X is a compact, convex set with interior points, and the norm · is induced by the o-symmetric convex body B X , called the unit ball of X, via x = min{λ ≥ 0 : x ∈ λB X }, ∀x ∈ R n .

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Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

Cited by 19 publications

References 34 publications

Is a complete, reduced set necessarily of constant width?

Is a complete, reduced set necessarily of constant width?

Geometry of Simplices in Minkowski Spaces

Complete sets need not be reduced in Minkowski spaces

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