On local convexity of nonlinear mappings between Banach spaces
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AbstractWe find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
1971 ◽
Vol 14
(1)
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pp. 119-120
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1976 ◽
Vol 17
(2)
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pp. 89-97
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2011 ◽
Vol 84
(3)
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pp. 353-361
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