For a C1 function f:Rn →R (n≥2) , we consider the least number k of distinct critical points that f must possess when restricted to the sphere S={x∈Rn :∥x∥=1} . Clearly k≥2 (for f attains its absolute minimum and maximum on S), and a result of Lyusternik and Shnirelʹman establishes that k=n if f is even. Here we prove that k=n if, for a given orthonormal system (ei ), max S∩Vi f
Chiappinelli, R. (2000). On the number of critical points of a C1 function on the sphere. GLASGOW MATHEMATICAL JOURNAL, 42(2), 283-287 [10.1017/S0017089500020152].
On the number of critical points of a C1 function on the sphere
CHIAPPINELLI, RAFFAELE
2000-01-01
Abstract
For a C1 function f:Rn →R (n≥2) , we consider the least number k of distinct critical points that f must possess when restricted to the sphere S={x∈Rn :∥x∥=1} . Clearly k≥2 (for f attains its absolute minimum and maximum on S), and a result of Lyusternik and Shnirelʹman establishes that k=n if f is even. Here we prove that k=n if, for a given orthonormal system (ei ), max S∩Vi fI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/11365/32307
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