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On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky (2012)

Open Mathematics

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.

On monotone minimal cuscos

Karel Pastor, Dušan Bednařík (2001)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

On the characterization of some families of closed convex sets.

Goberna, Miguel A., Jornet, Valentin, Rodriguez, Margarita (2002)

Beiträge zur Algebra und Geometrie

On the continuity of functions.

Almeida, Ricardo (2007)

APPS. Applied Sciences

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$ -dimensional Euclidean space $ℝ^{n}$ . It is proved that if $n \geq 2$ , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $G L (n)$ covariant, and associative if and only if it is $L_{p}$ addition for some $1 \leq p \leq \infty$ . It is also demonstrated that if $n \geq 2$ ,...

Outer $γ$ -convex functions on a normed space.

Phan Thanh An (2005)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]