On a Conjecture of a . J . Hoffman
نویسندگان
چکیده
The following question was recently raised by A. J. Hoffman [2]: "If P is a d-polytope, i>0 an integer and &,••*, C* closed convex sets in P, such that every /-flat that meets P meets Uf=i d; do there exist polytopes Di, • • • , Dk, with DiQd for all 1 ̂ i^k, such that if a ¿-flat meets P, it meets Uf_x 7>,?" The purpose of this note is to give a counterexample to this conjecture, with d = 3, t = 1 and k = 4 (see Remarks 1 and 2). A d-polytope here means the convex hull of a set of finitely many points in the Euclidean d-space Ed, having a nonempty interior, see [1]. Let P be the poly tope on the following six vertices in E3:Ai = (0, 0, 0), A* = (3, 0, 0), A3 = i0, 3, 0), ¿«=(0, 0, 20), ¿. = (3, 0, 20) and^6=(0, 3, 20). P is a prism with a base Bi at the level 2 = 0 and a base B2 at the level z = 20, where
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