Three Problems About 4-Polytopes

摘要: To some extent, we can claim to“understand” 3-dimensional polytopes. in fact, Steinitz’ Theorem “the combinatorial types of 3-polytopes are given by the simple, 3-connected planar graphs” (Steinitz, see Steinitz & Rademacher [12]) reduces much geometry to entirely combinational questions. Its powerful extensions answer basic questions about representing actual polytopes: “every 3-polytope be realized with rational vertex coordiantes”(a trivial consequence inductive proof for theorem), “every type shape one facet (2-face)arbitrarily prescribed” (a theorem obtained subtle adaption proof, BArnette Grýbaum [2]), “the space all realizations a convex 3-polytope, up affine equivalence, is contractible, and thus particular connected” (this what actually proved, see[12]).