Gel fand, Zelevinskiȋ, and Kapranov showed that the regular triangulations of a "primary" convex polytope can be viewed as the vertices of another convex polytope, which is said to be secondary. Billera, Filliman, and Sturmfels gave a geometric construction of the secondary polytope, based upon Gale transforms. We apply this construction to describing regular triangulations of nonconvex polytopes. We also discuss the problem of triangulating nonconvex polytopes with Steiner points.2000 Mathematics Subject Classification. Primary 52B11; Secondary 52B35, 68U05. 673License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 674 M. YU. ZVAGEL SKIȊ, A. V. PROSKURNIKOV, AND YU. R. ROMANOVSKIȊ that the optimal (in the number of extra points) regular subdivision of each minimal nonregular configuration S requires only one Steiner point, which can be put on any of the edges in S. In Theorem 2, we give a rigorous mathematical proof of this conjecture if S contains only three edges. In the proof, we explicitly describe the interval in which the Steiner point should be inserted. At the end of §4, we give an example of a minimal nonregular configuration, which contains arbitrarily many edges and admits a one-point regular subdivision.Of course, the regularity of the 1-skeleton does not imply the regular triangulability of the entire polytope P . However, it turns out that the regularity of its (d − 2)-skeleton guarantees the triangulability of P . This fact is a direct consequence of Shewchuk's results (see [5,6]). Usually, the Shewchuk triangulation is not regular, but in some natural sense it is close to a regular triangulation of the (d − 2)-skeleton. A useful link between Shewchuk's results and our approach is discussed at the end of §3.The authors are grateful to S. V. Duzhin for proposing a study of this subject in the computational geometry project headed by him, and also for his attention to this work and for discussing the results. We regard as our pleasant duty to thank N. E. Mnëv, who directed our attention to the theory of secondary polytopes and provided us with copies of several articles published in this area. We also thank K. V. Vyatkina, who informed us of the recent paper [6] by Shewchuk.
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