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--- tutorial:face_lattice_tutorial [2012/02/02 10:56] – large polytopes joswig
+++ tutorial:face_lattice_tutorial [2012/05/22 11:57] – [Dealing with Large Polytopes] joswig
@@ Line 60: / Line 60: @@
 In general, there is no way to tell ahead of time which convex hull algorithm works best.  So, for your own experiments you will have to try.  To get an idea you might want to look up:
-  * Avis, David; Bremner, David; Seidel, Raimund How good are convex hull algorithms? 11th ACM Symposium on Computational Geometry (Vancouver, BC, 1995). Comput. Geom. 7 (1997), no. 5-6, 265–301.
+  * Avis, David; Bremner, David; Seidel, Raimund: How good are convex hull algorithms? 11th ACM Symposium on Computational Geometry (Vancouver, BC, 1995). Comput. Geom. 7 (1997), no. 5-6, 265–301.
-  * Joswig, Michael Beneath-and-beyond revisited. Algebra, geometry, and software systems, 1–21, Springer, Berlin, 2003.
+  * Joswig, Michael: Beneath-and-beyond revisited. Algebra, geometry, and software systems, 1–21, Springer, Berlin, 2003.
 The subsequent second stage looks as above; but the difference is that VERTICES_IN_FACETS is known already.
@@ Line 70: / Line 70: @@
 </code>
 The executive summary: While polymake is designed to to all kinds of things automatically, you might have to guide it a little if you are computing with large or special input.
+One more caveat:  A //d//-polytope with //n// vertices has at most //O(n^(d/2))// facets.  This is the consequence of the Upper-Bound-Theorem.
+  * McMullen, Peter:  The maximum numbers of faces of a convex polytope. Mathematika 17 (1970) 179-184.
+This number is actually attained by neighborly polytopes; for example, by the cyclic polytopes.