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Monotone Valuations on the Space of Convex Functions

100%

Cavallina L., Colesanti A.

Analysis and Geometry in Metric Spaces

2015

tom 3

nr 1

We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.

On local convexity of nonlinear mappings between Banach spaces

84%

Banakh I., Banakh T., Plichko A., Prykarpatsky A.

Open Mathematics

2012

tom 10

nr 6

2264-2271

We 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.

Ograniczanie wyników

1 Analysis and Geometry in Metric Spaces

1 Open Mathematics

1 Banakh I.

1 Banakh T.

1 Cavallina L.

1 Colesanti A.

1 Plichko A.

1 Prykarpatsky A.

1 2015

1 2012

Wyniki wyszukiwania

Monotone Valuations on the Space of Convex Functions

On local convexity of nonlinear mappings between Banach spaces