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Wetzel, John E. Knox, Steven Wayne 2011-05-07T13:13:10Z 2011-05-07T13:13:10Z 10000-01-01 1996 Let $\pi$ 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 $f\sb{d-2}(\pi(P))$ of $\pi(P)$:$$\eqalign{f\sb2(P)&\ge{1\over2}f\sb1(\pi(P))+2\ {\rm if}\ d=3,\cr f\sb{d-1}(P)&\ge2{\sqrt{f\sb{d-2}(\pi(P))}}\ {\rm if}\ d\ge4.\cr}$$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 $d\ge4.$ Made available in DSpace on 2011-05-07T13:13:10Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9702563.pdf: 5529285 bytes, checksum: b7f5692ed35ae3aca8daf3f590fa4b3e (MD5) Previous issue date: 1996 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:51:48Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:23:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only 9780591088069 AAI9702563 (UMI)AAI9702563 http://hdl.handle.net/2142/21590 eng Copyright 1996 Knox, Steven Wayne Mathematics The number of facets of a projection of a convex polytope text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.