Do plane projections determine a convex polytope? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:29:24Z http://mathoverflow.net/feeds/question/15612 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15612/do-plane-projections-determine-a-convex-polytope Do plane projections determine a convex polytope? Igor Pak 2010-02-17T19:18:47Z 2013-02-13T17:15:34Z <p>Suppose a compact convex body \$P \subset \Bbb R^3\$ has only polygonal orthogonal projections onto a plane. Does this imply that \$P\$ is a convex polytope?</p> <p>This question occurred to me when I was making exercises for <a href="http://www.math.ucla.edu/~pak/book.htm" rel="nofollow">my book</a>. I figured this is probably easy and well known, but the literature hasn't been any help. One remark: if the number of sides of all polygons is bounded by \$n\$, the problem might be easier. Furthermore, if \$P\$ is <em>assumed</em> to be a convex polytope, <a href="http://www.cs.princeton.edu/~chazelle/pubs/ComplexityCuttingComplexes.pdf" rel="nofollow">this</a> elegant paper by Chazelle-Edelsbrunner-Guibas (1989) gives a (perhaps, unexpectedly large) sharp \$\exp O(n \log n)\$ upper bound on the number of vertices of \$P\$ (ht Csaba Toth who <a href="http://math.ucalgary.ca/~cdtoth/cx-stabbing-rev.pdf" rel="nofollow">generalized</a> this to higher dimensions). </p> http://mathoverflow.net/questions/15612/do-plane-projections-determine-a-convex-polytope/15815#15815 Answer by Mark Meckes for Do plane projections determine a convex polytope? Mark Meckes 2010-02-19T14:34:17Z 2010-02-19T15:41:24Z <p>Theorem 4.1 of <a href="http://www.ams.org/mathscinet-getitem?mr=105651" rel="nofollow">this paper by Klee</a> says yes. Moreover, the result generalizes to higher dimensions for projections of arbitrary dimension \$\ge 2\$.</p> http://mathoverflow.net/questions/15612/do-plane-projections-determine-a-convex-polytope/121729#121729 Answer by Anton Petrunin for Do plane projections determine a convex polytope? Anton Petrunin 2013-02-13T17:15:34Z 2013-02-13T17:15:34Z <p>Here is a more direct proof of this statement.</p> <p><a href="http://arxiv.org/pdf/1302.2354.pdf" rel="nofollow">A Short Proof of Klee's Theorem</a> by John J. Zanazzi</p>