The Lower Bound Theorem for polytopes that approximate C1-convex bodies

Karim Adiprasito; José Alejandro Samper

The Lower Bound Theorem for polytopes that approximate C¹-convex bodies

Research output: Contribution to journal › Conference article › peer-review

Abstract

The face numbers of simplicial polytopes that approximate C¹-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence {Pn}⁸_n=0 of simplicial polytopes converges to a C¹-convex body in the Hausdorff distance, then the entries of the g-vector of Pn converge to infinity.

Original language	English (US)
Pages (from-to)	277-287
Number of pages	11
Journal	Discrete Mathematics and Theoretical Computer Science
State	Published - 2014
Externally published	Yes
Event	26th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2014 - Chicago, United States Duration: Jun 29 2014 → Jul 3 2014

Keywords

Approximation theory
Convex bodies
F-vector theory
Geometric Combinatorics
Lower bound theorem
Polytopes

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)
Discrete Mathematics and Combinatorics

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title = "The Lower Bound Theorem for polytopes that approximate C1-convex bodies",

abstract = "The face numbers of simplicial polytopes that approximate C1-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence {Pn}8n=0 of simplicial polytopes converges to a C1-convex body in the Hausdorff distance, then the entries of the g-vector of Pn converge to infinity.",

keywords = "Approximation theory, Convex bodies, F-vector theory, Geometric Combinatorics, Lower bound theorem, Polytopes",

author = "Karim Adiprasito and Samper, {Jos{\'e} Alejandro}",

note = "Funding Information: ∗Email: adiprasito@ihes.fr. K. Adiprasito has been supported by an EPDI postdoctoral fellowship and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533. †Email: samper@math.washington.edu. Jose Samper{\textquoteright}s research is partially supported by a graduate fellowship from NSF Grant DMS-1069298 Publisher Copyright: {\textcopyright} 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France Copyright: Copyright 2020 Elsevier B.V., All rights reserved.; 26th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2014 ; Conference date: 29-06-2014 Through 03-07-2014",

year = "2014",

language = "English (US)",

pages = "277--287",

journal = "Discrete Mathematics and Theoretical Computer Science",

issn = "1365-8050",

publisher = "Maison de l'informatique et des mathematiques discretes",

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T1 - The Lower Bound Theorem for polytopes that approximate C1-convex bodies

AU - Adiprasito, Karim

AU - Samper, José Alejandro

N1 - Funding Information: ∗Email: adiprasito@ihes.fr. K. Adiprasito has been supported by an EPDI postdoctoral fellowship and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533. †Email: samper@math.washington.edu. Jose Samper’s research is partially supported by a graduate fellowship from NSF Grant DMS-1069298 Publisher Copyright: © 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2014

Y1 - 2014

N2 - The face numbers of simplicial polytopes that approximate C1-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence {Pn}8n=0 of simplicial polytopes converges to a C1-convex body in the Hausdorff distance, then the entries of the g-vector of Pn converge to infinity.

AB - The face numbers of simplicial polytopes that approximate C1-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence {Pn}8n=0 of simplicial polytopes converges to a C1-convex body in the Hausdorff distance, then the entries of the g-vector of Pn converge to infinity.

KW - Approximation theory

KW - Convex bodies

KW - F-vector theory

KW - Geometric Combinatorics

KW - Lower bound theorem

KW - Polytopes

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M3 - Conference article

AN - SCOPUS:85082971153

SP - 277

EP - 287

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

T2 - 26th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2014

Y2 - 29 June 2014 through 3 July 2014