Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.eng.math.msu.su/download/theses1.pdf Дата изменения: Mon Sep 29 04:23:45 2014 Дата индексирования: Sat Apr 9 21:56:42 2016 Кодировка:

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, English Department On Borsuk's problem by A. Lukina, group 305, English teacher: A.A.Savchenko

1. The problem under consideration is an open question in combinatorial geometry, the branch of mathematics that appeared only in 19th century. It studies combinatorial properties of finite or discrete bodies or bodies with some particular characteristics, for instance, of a certain diameter. 2. Borsuk suggested that every set of diameter one in Rd can be partitioned into d+1 pieces of diameter smaller than one. 3. Kahn and Kalai proved that the conjecture does not hold for every d. So it is quite interesting to understand for which d and for which kinds of bodies Borsuk was right. 4. A number of questions arise from this problem. For instance, we can partition bodies not only into d+1 pieces, but into some number of sets of smaller diameter. 5. It is useful to deal with not just a body of diameter one, but with its approximation, called universal cover system. The better approximation is, the better bounds can be obtained. 6. There are some common ways to amend universal covers. One of them was used by GrЭnbaum to prove Borsuk's conjecture for d=3. 7. A number of results are known in R2, but there is almost nothing in R3, so new interesting relations can be discovered without using complex techniques.

English teacher: __________________/ A.A.Savchenko