Dedicated to Professor Otto Haupt with best wishes on his lOOth birthday.Abstract. We show that an isomorphism between the graphs of two simple polytopes of arbitrary dimension can always be extended to an isomorphism between the polytopes themselves. It has been convenient to study the dual situation, involving what we like to call the puzzle of a simplicial polytope.

An old problem in combinatorial geometry is to determine when one or more measurable sets in R d admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257-1261]. A related topological problem is the question of (non)existence of a map f : (S d ) k → S(U ), equivariant with respect to the Weyl group W k = B k := (Z/2) ⊕k S k , where U is a representation of W k and S(U ) ⊂ U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R 8 [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167]. The obstruction in this case is identified as the element 2X ab ∈ H 1 (D 8 ; Z) ∼ = Z/4, where X ab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel-Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and BourginYang theorems, Ergodic Theory Dynam. Systems 8 * (1988) 73-85].

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We give an easy general construction for uniform oriented matroids with disconnected realization space. This disproves the longstanding isotopy conjecture for simple line arrangements or order types in the plane.We write ~(M) for the space of all vector realizations (x~,..., xn) 6 (R3) n of a rank 3 oriented matroid M on n points. (In other words, ~(M) is the set of 3 × n-matrices whose maximal minors have signs given by the alternating map M: {1, 2,...,/./}3_~ {_, 0, +}.) If M is uniform (i.e., all minors are nonzero) then ~(M) is an open subset of R 3". White's earlier paper [8] gives a nonuniform oriented matroid Mw with ~(Mw) disconnected and n =42. The aim of the present paper is to present a uniform oriented matroid ~/ with .~(hT/) disconnected. That is, /~ does not have the isotopy property.A rank 3 oriented matroid M is said to be constructible if (x~, x2, x3, x4) is

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