On the Reconstruction of Polytopes

Joseph Doolittle; Eran Nevo; Guillermo Pineda-Villavicencio; Julien Ugon; David Yost

doi:10.1007/s00454-018-9997-9

On the Reconstruction of Polytopes

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio^*, Julien Ugon, David Yost

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.

Original language	American English
Pages (from-to)	285-302
Number of pages	18
Journal	Discrete and Computational Geometry
Volume	61
Issue number	2
DOIs	https://doi.org/10.1007/s00454-018-9997-9
State	Published - 15 Mar 2019

Bibliographical note

Funding Information:
Research of Nevo was partially supported by Israel Science Foundation Grant ISF-1695/15. Research of Pineda-Villavicencio was partly supported by the Indonesian government Scheme P3MI, Grant No. 1016/I1.C01/PL/2017. Research of Ugon was supported by ARC discovery Project DP180100602.

Funding Information:
We thank Micha Perles for helpful discussions and the referees for many valuable comments and suggestions. Guillermo Pineda would like to thank Michael Joswig for the hospitality at the Technical University of Berlin and for many fruitful discussions on the topics of this research. Joseph Doolittle would like to thank Margaret Bayer for pushing for more results and keeping the direction of exploration straight.

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Reconstruction
Simple polytope
k-Skeleton

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10.1007/s00454-018-9997-9

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title = "On the Reconstruction of Polytopes",

abstract = "Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.",

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note = "Funding Information: Research of Nevo was partially supported by Israel Science Foundation Grant ISF-1695/15. Research of Pineda-Villavicencio was partly supported by the Indonesian government Scheme P3MI, Grant No. 1016/I1.C01/PL/2017. Research of Ugon was supported by ARC discovery Project DP180100602. Funding Information: We thank Micha Perles for helpful discussions and the referees for many valuable comments and suggestions. Guillermo Pineda would like to thank Michael Joswig for the hospitality at the Technical University of Berlin and for many fruitful discussions on the topics of this research. Joseph Doolittle would like to thank Margaret Bayer for pushing for more results and keeping the direction of exploration straight. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

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N2 - Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.

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On the Reconstruction of Polytopes

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Keywords

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