The number of facets of a projection of a convex polytope

Let π be orthogonal projection of $\IR\sp{d}$ onto a hyperplane and let P be a d-polytope in $\IR\sp{d}$. The following relations hold on the numbers of facets $f\sb{d-1}(P)$ of P and f\sb{d-2}(π(P)) of π(P):$\eqalign{f\sb2(P)&\ge{1\over2}f\sb1(π(P))+2 {\rm if} d=3,\cr f\sb{d-1}(P)&\ge2{\sqrt{f\sb{d-2}(π(P))}} {\rm if} d\ge4.\cr}$Both bounds are sharp. If d = 3 the range of the map $P\mapsto (f\sb2(P), f\sb1(\pi(P)))$ is $\{(x,y)\in{\rm I\!N}\sp2:x\ge{1\over2}y+2,y\ge3\}.$ If d = 4 the range of the map $P\mapsto (f\sb3(P),f\sb2(\pi(P)))$ is $\{(x, y) \in{\rm I\!N}\sp2:x\ge 2\sqrt y, x\ge 5,y\ge 4\}.$ For each $d\ge 4$ and each integer $n\ge 3$ there is a d-polytope $P\sb{d}(n)$ with ($f\sb{d-1}(P\sb{d}(n)), f\sb{d-2}(\pi(P\sb{d}(n))))=(2\sp{d-3}n, 2\sp{2(d-4)}n\sp2).$ Thus for each fixed value of d the infimum of the ratio ${f\sb{d-1}(P)}\over{f\sb{d-2}(\pi(P))}$ as P varies over all d-polytopes, is $1\over2$ if d = 3 and 0 if $d\ge4.$

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Knox, Steven Wayne

Contributors dc:contributor

• Wetzel, John E.

• Mathematics

Statement dc:rights

• Copyright 1996 Knox, Steven Wayne

Language dc:language

eng

Identifier

9780591088069
AAI9702563
(UMI)AAI9702563

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/21590