Strongly Midquasiconvex Functions

Let $V$ be a nonempty convex subset of a normed space $X$ and let $\e>0$ and $p>0$ be given. A function $f: V \to \R$ is called $(\e,p)$-strongly midquasiconvex if $$f(\frac{x+y}{2}) \leq \max [f(x), f(y)]-\e(\frac{\|x-y\|}{2})^p \text{\ \ for\ \ } x,y \in V.$$ We call $f$ $p$-strongly midquasiconvex if it is $(\e,p)$-strongly midquasiconvex with a certain $\e>0$. We show that if either $p<1$ and $\dim V=1$ or $p<2$ and $\dim V>1$ then there are no $p$-strongly midquasiconvex functions defined on $V$. On the other hand if $X$ is an inner product space with $\dim X \geq 2$, $p \geq 2$, then there exists an $(1,p)$-strongly midquasiconvex function defined on an arbitrary ball in $X$. Consequently, the case when $p=1$ and $\dim V=1$ is of a special interest. Under this assumptions we characterize lower semicontinuous $1$-strongly midquasiconvex functions.