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Rigidity, tensegrity, and reconstruction of polytopes under metric constraints
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Winter, Martin (2023) Rigidity, tensegrity, and reconstruction of polytopes under metric constraints. International Mathematics Research Notices . rnad298. doi:10.1093/imrn/rnad298 ISSN 1073-7928. (In Press)
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Official URL: https://doi.org/10.1093/imrn/rnad298
Abstract
We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset {\mathbb {R}}^{d}$ and $Q\subset {\mathbb {R}}^{e}$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack these questions and we verify them in three relevant special cases: $P$ and $Q$ are centrally symmetric, $Q$ is a slight perturbation of $P$, and $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q:V(G_{P})\to {\mathbb {R}}^{e}$ of the edge-graph $G_{P}$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
SWORD Depositor: | Library Publications Router | ||||||
Library of Congress Subject Headings (LCSH): | Convex polytopes, Reconstruction (Graph theory) | ||||||
Journal or Publication Title: | International Mathematics Research Notices | ||||||
Publisher: | Oxford University Press | ||||||
ISSN: | 1073-7928 | ||||||
Official Date: | 21 December 2023 | ||||||
Dates: |
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Article Number: | rnad298 | ||||||
DOI: | 10.1093/imrn/rnad298 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | In Press | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 28 March 2024 | ||||||
Date of first compliant Open Access: | 28 March 2024 | ||||||
RIOXX Funder/Project Grant: |
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