Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-04T21:16:29.206Z Has data issue: false hasContentIssue false

8 - Faces

Published online by Cambridge University Press:  14 March 2024

Guillermo Pineda Villavicencio
Affiliation:
Deakin University, Victoria
Get access

Summary

We investigate numbers of faces of polytopes. We begin with the face numbers of 3-polytopes. The characterisation of $f$-vectors of $d$-polytopes ($d\ge 4$) is beyond our current means.In view of this, researchers have considered characterisations of the "projections" of the $f$-vectors, namely the proper subsequences of the $f$-vector; we review the existing results. Section 8.2 gives a proof of a theorem of Xue (2021) on the minimum number of faces of $d$-polytopes with at most $2d$ vertices, answering a conjecture of Grunbaum (2003). This is followed by results on the minimum number of faces of $d$-polytopes with more than $2d$ vertices. We then discuss the lower and upper bound theorems for simplicial polytopes, due to Barnette (1973) and McMullen (1970), respectively, and their extensions such as the $g$-conjecture of McMullen (1971), now the $g$-theorem. The proof of the lower bound theorem connects rigidity theory and the combinatorics polytopes. The chapter ends with a discussion of the flag vector of a polytope. This includes a result of Bayer and Billera (1985) on linear equations for flag vectors like the Dehn--Sommerville’s equations for simplicial polytopes.

Type
Chapter
Information
Polytopes and Graphs , pp. 340 - 393
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Faces
  • Guillermo Pineda Villavicencio, Deakin University, Victoria
  • Book: Polytopes and Graphs
  • Online publication: 14 March 2024
  • Chapter DOI: https://doi.org/10.1017/9781009257794.009
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Faces
  • Guillermo Pineda Villavicencio, Deakin University, Victoria
  • Book: Polytopes and Graphs
  • Online publication: 14 March 2024
  • Chapter DOI: https://doi.org/10.1017/9781009257794.009
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Faces
  • Guillermo Pineda Villavicencio, Deakin University, Victoria
  • Book: Polytopes and Graphs
  • Online publication: 14 March 2024
  • Chapter DOI: https://doi.org/10.1017/9781009257794.009
Available formats
×