In 1832, Jakob Steiner (Systematische Entwicklung der Abh?ngigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially equivalent to polyhedra inscribed in the sphere. This question was answered in the 1990s by Igor Rivin ( Ann. of Math. 143(1996), 51–70), as a byproduct of his classification of ideal polyhedra in hyperbolic 3-space. Rivin also proposed a more direct approach to these results in Rivin (Ann. of Math. 139 (1994), 553–580). In this paper, we prove a combinatorial result (Theorem 6) which enables one to complete the program of Rivin (Ann. Math. 139(1994), 553–580).
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机译:1832年,雅各布·施泰纳(Jakob Steiner)(柏林雷默(Reimer)的系统几何学专家Gestalten von einander)要求对这些平面图进行刻画,这些平面图在组合上等同于球体中刻画的多面体。这个问题在1990年代由Igor Rivin(数学年鉴143(1996),第51-70页)回答,是他在双曲3空间中理想多面体分类的副产品。 Rivin在Rivin中也对这些结果提出了一种更直接的方法(数学年鉴139(1994),553-580)。在本文中,我们证明了组合结果(定理6),该结果使一个人可以完成Rivin的程序(Ann。Math。139(1994),553-580)。
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