Relationships Among Flag f -Vector Inequalities for Polytopes

We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 31; no. 2; pp. 257 - 273
Main Author Stenson, Catherine
Format Journal Article
LanguageEnglish
Published New York Springer Nature B.V 01.02.2004
Subjects
Algebra
Geometry
Mathematical analysis
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Summary:We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of inequalities is the {\bf cd}-index; among all $d$-polytopes, each {\bf cd}-index coefficient is minimized on the $d$-simplex. We show that not all of the {\bf cd}-index inequalities are implied by the toric $g$-vector inequalities, and that not all of the toric $g$-vector inequalities are implied by the {\bf cd}-index inequalities. Finally, we show that some inequalities from convolutions of {\bf cd}-index coefficients are implied by other {\bf cd}-index inequalities. [PUBLICATION ABSTRACT]
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-003-0756-0