{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21590"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21590","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The number of facets of a projection of a convex polytope","abstract":"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 $d\\ge4.$","abstract_html":"Let <span class=\"etd-inline-math\">&pi;</span> 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 <span class=\"etd-inline-math\">f\\sb{d-2}(&pi;(P))</span> of <span class=\"etd-inline-math\">&pi;(P)</span>:$<span class=\"etd-inline-math\">\\eqalign{f\\sb2(P)&amp;\\ge{1\\over2}f\\sb1(&pi;(P))+2 {\\rm if} d=3,\\cr f\\sb{d-1}(P)&amp;\\ge2{\\sqrt{f\\sb{d-2}(&pi;(P))}} {\\rm if} d\\ge4.\\cr}</span>$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 $d\\ge4.$","abstract_has_math":true,"creators":["Knox, Steven Wayne"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Wetzel, John E."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:13:10Z","date_published":"2011-05-07T13:13:10Z","updated_at":"2026-07-22T22:25:18Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1996 Knox, Steven Wayne"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591088069","AAI9702563","(UMI)AAI9702563"],"render_values":[{"text":"9780591088069","href":null,"code":true},{"text":"AAI9702563","href":null,"code":true},{"text":"(UMI)AAI9702563","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21590","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Wetzel, John E."]},{"key":"dc:creator","label":"Author","values":["Knox, Steven Wayne"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:13:10Z","10000-01-01","1996"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1996 Knox, Steven Wayne"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591088069","AAI9702563","(UMI)AAI9702563","http://hdl.handle.net/2142/21590"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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"]},{"key":"dc:title","label":"Title","values":["The number of facets of a projection of a convex polytope"]}]}],"canonical_facts":{"dc:contributor":["Wetzel, John E."],"dc:creator":["Knox, Steven Wayne"],"dc:date":["2011-05-07T13:13:10Z","10000-01-01","1996"],"dc:description":["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"],"dc:identifier":["9780591088069","AAI9702563","(UMI)AAI9702563","http://hdl.handle.net/2142/21590"],"dc:language":["eng"],"dc:rights":["Copyright 1996 Knox, Steven Wayne"],"dc:subject":["Mathematics"],"dc:title":["The number of facets of a projection of a convex polytope"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:18Z"}