Intersection bodies of polytopes

Berlow, Katalin; Brandenburg, Marie-Charlotte; Meroni, Chiara; Shankar, Isabelle

doi:10.1007/s13366-022-00621-7

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…The goal of this article is to investigate the behavior of intersection bodies of polytopes under translations, and to determine under which translations the intersection body is convex. In our previous work [BBMS22] we exhibit rich semialgebraic structures of intersection bodies of polytopes. However, in general, the intersection body IP of a polytope P is not a basic semialgebraic set, and there exists a central hyperplane arrangement which describes the regions in which the topological boundary of IP is defined by a fixed polynomial.…”

Section: Introductionmentioning

confidence: 99%

“…The article is structured as follows. In Section 2 we review the main concepts and notation from [BBMS22]. In Section 3 we introduce an affine hyperplane arrangement and describe how it governs the behavior of IP under translation of P .…”

Section: Introductionmentioning

confidence: 99%

“…PreliminariesWe will rely on methods and results which were developed in[BBMS22]. In this section we review the most important concepts and results we are going to make use of.…”

mentioning

confidence: 99%

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Intersection Bodies of Polytopes: Translations and Convexity

Brandenburg¹,

Meroni²

2023

Preprint

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We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P . We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of I(P + t) can be extended to polynomials in variables t ∈ R d within each region of the arrangement. Establishing the convexity space as the set of translations such that I(P + t) is convex, we fully characterize it for two-dimensional polytopes and partially characterize it for higher dimensions, revealing unexpected finite behavior in the two-dimensional case and for the d-dimensional cube.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intersection Bodies of Polytopes: Translations and Convexity

Brandenburg¹,

Meroni²

2023

Preprint

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Intersection bodies of polytopes: translations and convexity

Brandenburg,

Meroni

2024

J Algebr Comb

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We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of $$I(P+t)$$ I ( P + t ) can be extended to polynomials in variables $$t\in \mathbb {R}^d$$ t ∈ R d within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions.

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