Collapsing the medial axis of a polytope - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:29:13Z http://mathoverflow.net/feeds/question/10031 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10031/collapsing-the-medial-axis-of-a-polytope Collapsing the medial axis of a polytope Shinpei 2009-12-29T12:50:10Z 2009-12-29T13:06:56Z <p><strong>Let X be a convex polyhedron in hyperbolic 3-space.</strong></p> <p><strong>Let M be the medial axis of X.</strong></p> <p><strong>Question: Is M collapsible?</strong></p> <p>It is easy to see that M is contractable. In the case of Euclidian 3-space, instead of hyperbolic 3-space, I think I have an elementary proof for the analogues statement. In hyperbolic space, I guess that the statement still holds, but I do not have a proof. </p> <p>I would appreciate comments and references for the above question.</p> <p><strong>Definitions for the above question</strong></p> <p>The MEDIAL AIXS of X : consider a maximal round ball inscribed in X (maximal with respect to set inclusion), which at least has two points of tangency (on $\partial X$). Take the union of the centers of all such maximal balls inscribed in X. Then, this union is called the medial axis of X, and it is a polygonal complex, to be precise, after adding the 1-skeleton of the boundary of X.</p> <p>M is COLLAPSIBLE: There is a strong deformation retract of M to a point that is a composition of certain simplicial homotopies, each of which reduces a single cell.</p>