View source for Lyusternik-Shnirel'man-Borsuk covering theorem

A theorem usually stated as follows:

1) Each closed covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202001.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202002.png" /> contains at least one set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202003.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202004.png" />.

Contrary to the equivalent [[Borsuk–Ulam theorem|Borsuk–Ulam theorem]], it seems to be not common to use the same name also for the following equivalent symmetric versions:

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202005.png" /> be closed sets with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202006.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202007.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202008.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l1202009.png" />.

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l12020010.png" /> [[#References|[a2]]].

In all these results, the estimates are optimal (in 3), in fact, equality holds). It is worth mentioning that 2) gave the motivation for the notion of the genus of a set symmetric with respect to a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l12020011.png" />-action.

For other equivalent versions and for generalizations to coverings involving other symmetries (e.g. with respect to free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l12020012.png" />-actions), cf. [[#References|[a3]]] and the references therein.

One major field of applications are estimates of the number of critical points of even functionals; this can be used, e.g., in the theory of differential equations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Borsuk,   "Drei Sätze über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120200/l12020013.png" />—dimensionale Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Lyusternik,   L. Shnirel'man,   "Topological methods in variational problems" , Issl. Inst. Mat. Mekh. OMGU  (1930)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Steinlein,   "Spheres and symmetry: Borsuk's antipodal theorem"  ''Topol. Methods Nonlinear Anal.'' , '''1'''  (1993)  pp. 15–33</TD></TR></table>