Convexity and openness with linear rate (Q1281553)

Let \(X\), \(Y\) be Banach spaces, \(B\) denotes the respective closed unit ball and \(x+\tau B\) the closed ball with radius \(\tau\) around \(x\). A cone in \(X\) is a nonempty subset \(K\subset X\) that contains with an \(x\in X\) all multiples \(\alpha x\) for \(\alpha\geq 0\). A cone \(K\) is said to be normal if there exists a number \(\delta> 0\) such that \(x_1,x_2\in K\), \(\| x_1\|= \| x_2\|=1\) implies \(\| x_1+ x_2\|\geq\delta\). A mapping \(F:X\to Y\) is said to be open with linear rate around \(x_0\in X\) if then exist numbers \(\tau_0>0\) and \(c>0\) such that for each \(x\in x_0+\tau_0 B\) and for each \(\pi\in [0,\tau_0]\), \(F(x)+ \tau B\subset F(x+ (\tau B))\) holds. The aim of the present paper is to supply conditions for openness with linear rate under certain convexity hypotheses on the mapping. The main result, Theorem 1 in Section 2, states that each convex continuous function on a Banach space is open with linear rate around each point that is not a minimum point of this function. For cone-convex and continuous mappings between two Banach spaces, where the image space is finite-dimensional and ordered by a normal cone, a sufficient condition for openness with linear rate is deduced in Section 3. In Section 4 it is shown that if, in addition, the mapping is Fréchet-differentiable at the considered point then this condition is also necessary, and the derivative is surjective. In Section 5, a Lyusternik-type theorem [\textit{L. A. Lyusternik}, Rec. Math. Moscou 41, 390-401 (1934; Zbl 0011.07401)] for mappings that are open with linear rate is proved and applied to convex functions and cone-convex mappings.