Dropping a vertex or a facet from a convex polytope

Shlomo Reisner; Carsten Schütt; Elisabeth Werner

doi:10.1515/form.2001.012

Dropping a vertex or a facet from a convex polytope

Shlomo Reisner, Carsten Schütt, Elisabeth Werner

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

18 Scopus citations

Abstract

There exist positive constants c₀ and c₁ = c₁ (n) such that for every 0 < ε < 1/2 the following holds: Let P be a convex polytope in IRⁿ having N ≥ cⁿ₀/ε vertices x₁ , . . . , X_N. Then there exists a subset A ⊂ {1, . . . , N}, card(A) ≥ (1 -2ε)N, such that for all i ∈ A (formula presented) Also, if P is a convex polytope in IRⁿ having N ≥ cⁿ₀/ε facets. Let H⁺_i be the half space determined by the facet F_i, which contains P (i = 1, . . . , N). Then there exists a subset A ⊂ {1, . . . ,N}, card(A) ≥ (1 - 2ε)N, such that for all i ∈ A (formula presented).

Original language	English (US)
Pages (from-to)	359-378
Number of pages	20
Journal	Forum Mathematicum
Volume	13
Issue number	3
DOIs	https://doi.org/10.1515/form.2001.012
State	Published - 2001
Externally published	Yes

Bibliographical note

Funding Information:
y Part of this work was done at the Erwin Schrödinger Institute for Mathematical Physics in Vienna, when the three authors participated in the Special Semester on Analysis, organized there by the Functional Analysis Group of the University of Linz. z Shlomo Reisner has been partially supported by the NSF under grant DMS-9626749

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title = "Dropping a vertex or a facet from a convex polytope",

abstract = "There exist positive constants c0 and c1 = c1 (n) such that for every 0 < ε < 1/2 the following holds: Let P be a convex polytope in IRn having N ≥ cn0/ε vertices x1 , . . . , XN. Then there exists a subset A ⊂ {1, . . . , N}, card(A) ≥ (1 -2ε)N, such that for all i ∈ A (formula presented) Also, if P is a convex polytope in IRn having N ≥ cn0/ε facets. Let H+i be the half space determined by the facet Fi, which contains P (i = 1, . . . , N). Then there exists a subset A ⊂ {1, . . . ,N}, card(A) ≥ (1 - 2ε)N, such that for all i ∈ A (formula presented).",

author = "Shlomo Reisner and Carsten Sch{\"u}tt and Elisabeth Werner",

note = "Funding Information: y Part of this work was done at the Erwin Schr{\"o}dinger Institute for Mathematical Physics in Vienna, when the three authors participated in the Special Semester on Analysis, organized there by the Functional Analysis Group of the University of Linz. z Shlomo Reisner has been partially supported by the NSF under grant DMS-9626749",

year = "2001",

doi = "10.1515/form.2001.012",

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T1 - Dropping a vertex or a facet from a convex polytope

AU - Reisner, Shlomo

AU - Schütt, Carsten

AU - Werner, Elisabeth

N1 - Funding Information: y Part of this work was done at the Erwin Schrödinger Institute for Mathematical Physics in Vienna, when the three authors participated in the Special Semester on Analysis, organized there by the Functional Analysis Group of the University of Linz. z Shlomo Reisner has been partially supported by the NSF under grant DMS-9626749

PY - 2001

Y1 - 2001

N2 - There exist positive constants c0 and c1 = c1 (n) such that for every 0 < ε < 1/2 the following holds: Let P be a convex polytope in IRn having N ≥ cn0/ε vertices x1 , . . . , XN. Then there exists a subset A ⊂ {1, . . . , N}, card(A) ≥ (1 -2ε)N, such that for all i ∈ A (formula presented) Also, if P is a convex polytope in IRn having N ≥ cn0/ε facets. Let H+i be the half space determined by the facet Fi, which contains P (i = 1, . . . , N). Then there exists a subset A ⊂ {1, . . . ,N}, card(A) ≥ (1 - 2ε)N, such that for all i ∈ A (formula presented).

AB - There exist positive constants c0 and c1 = c1 (n) such that for every 0 < ε < 1/2 the following holds: Let P be a convex polytope in IRn having N ≥ cn0/ε vertices x1 , . . . , XN. Then there exists a subset A ⊂ {1, . . . , N}, card(A) ≥ (1 -2ε)N, such that for all i ∈ A (formula presented) Also, if P is a convex polytope in IRn having N ≥ cn0/ε facets. Let H+i be the half space determined by the facet Fi, which contains P (i = 1, . . . , N). Then there exists a subset A ⊂ {1, . . . ,N}, card(A) ≥ (1 - 2ε)N, such that for all i ∈ A (formula presented).

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EP - 378

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 3

ER -

Dropping a vertex or a facet from a convex polytope

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