Existence of a Polyhedron with Prescribed Development
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EXISTENCE OF A POLYHEDRON WITH PRESCRIBED DEVELOPMENT Yu. A. Volkov∗
UDC 514.172, 514.722
This article is the publication of Yurii Aleksandrovich Volkov’s (1930–1981) Ph.D. thesis, in which A. D. Aleksandrov’s famous theorem on the existence of a convex polyhedron with a given development is proved using a variational method. Bibliography: 5 titles.
Preface This article presents Yuri Alexandrovich Volkov’s (02.09.1930–23.03.1981) PhD thesis. He completed it at Leningrad State University under the supervision of A. D. Alexandrov and successfully defended it in 1955. In his thesis, Volkov used the variational method to prove the existence of a polyhedron with a pre-specified development (see [1, 2]). Unfortunately, thesis defence rules at the time did not require the results described in the thesis to be published, and as such Volkov’s proof was only known to the participants of Alexandrov’s geometry seminar. In 1956, Volkov presented his research on the Third USSR-wide Mathematical Congress, from which a small note was published in [5]. After his thesis defence, Volkov concentrated on the stability problem in Minkowski’s theorem, which he solved in 1959 ([4], p. 218). The proof of the existence of a cap with a pre-specified development was only published in 1960 ([6]); Volkov wrote that he would examine the general (convex polyhedron) case in his next article. As such, he changed his approach from the one he used for his PhD thesis (where he presents a proof of the polyhedron case first, then sketches the proof for the cap case). In the early 1960s, Volkov discovered that his method of abstract polyhedra yielded results in the Weyl–Cohn-Vossen problem (estimating the change in external distances on convex surfaces in response to changes in internal distances). In 1962, he presented his results on the Cohn-Vossen problem in a geometry seminar ([4], p. 223). The proof of Volkov’s bounds on the changes in distances did not rely on the proof of A. V. Pogorelov’s famous theorem stating that two closed isometric surfaces are equal, so Volkov proved that theorem as well. He had thus obtained by 1962 new proofs of the fundamental theorems of the theory of convex surfaces. But the results were not yet published, and he had to decide what to publish first. It is likely for this reason that Volkov’s PhD thesis results were of less importance for him. Volkov published his results on Minkowski’s problem [7] in 1963 and on the Cohn-Vossen problem [8, 9] in 1968 and conducted a brilliant defence of his doctorate thesis. Only in 1968, however, did he suggest using the method of abstract polyhedra to prove S. P. Olovianishnikov’s theorem on the existence of an infinite polyhedron [11] to graduate student Yelena Gavrilovnna Podgornova. She defended her PhD thesis, containing a detailed proof of Olovianishnikov’s theorem, in 1972. Based on Volkov and Podgornova’s research, the article [10] was written, in which Alexandrov and Olovianishnikov’s theorems are proved in full detail. This article was meant to be published in LSU’s Vestnik, b
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