Strongly midquasiconvex functions (Q2841080)

The paper is devoted to the investigation of the class of \((\epsilon,p)\)-strongly midquasiconvex functions. Given a nonempty convex subset \(V\) of a normed space \(X\), and \(\epsilon,p>0\), a function \(f:V\to R\) is said to be \((\epsilon,p)\)-strongly midquasiconvex if NEWLINE\[NEWLINE f\left(\frac{x+y}{2}\right)\leq \max[f(x),f(y)]-\epsilon \left(\frac{\|x-y\|}{2}\right)^p,\quad x,y\in V; NEWLINE\]NEWLINE if the inequality holds for every \(\epsilon>0\), then \(f\) is \(p\)-strongly midquasiconvex. The nonexistence of \(p\)-strongly midquasiconvex functions defined on \(V\) is related to the dimension \(\dim V\) of the affine space generated by \(V\); indeed, the authors prove that in the cases: \(p<1\) and \(\dim V=1\), or \(p<2\) and \(\dim V>1\), there are no \(p\)-strongly midquasiconvex functions defined on \(V\). In addition, if \(X\) is an inner product space with \(\dim X\geq 2\), \(p\geq 2\), then there exist \((1,p)\)-strongly midquasiconvex functions defined on any balls of \(X\). In the special case: \(p=1\) and \(\dim V=1\) they characterize lower semicontinuous \(1\)-strongly midquasiconvex functions.