Strongly Midquasiconvex Functions

Let

be a nonempty convex subset of a normed space

X X

and let

\varepsilon>0 ε > 0

and

p>0 p > 0

be given. A function

f: V \to {{\mathbb R}} f : V → R

is called

(\varepsilon,p) ( ε , p )

-strongly midquasiconvex if

f(\frac{x+y}{2}) \leq \max [f(x), f(y)]-\varepsilon(\frac{\|x-y\|}{2})^p \text{\ \ for\ \ } x,y \in V. f ( x + y 2 ) ≤ max ⁡ [ f ( x ) , f ( y ) ] − ε ( ∥ x − y ∥ 2 ) p   for   x , y ∈ V .

We call

f f

p p

-strongly midquasiconvex if it is

(\varepsilon,p) ( ε , p )

-strongly midquasiconvex with a certain

\varepsilon>0 ε > 0

. We show that if either

p<1 p < 1

and

\mathrm{dim\,\,}V=1 d i m     V = 1

or

p<2 p < 2

and

\mathrm{dim\,\,}V>1 d i m     V > 1

then there are no

p p

-strongly midquasiconvex functions defined on

V V

. On the other hand if

X X

is an inner product space with

\mathrm{dim\,\,}X \geq 2 d i m     X ≥ 2

,

p \geq 2 p ≥ 2

, then there exists an

(1,p) ( 1 , p )

-strongly midquasiconvex function defined on an arbitrary ball in

X X

. Consequently, the case when

p=1 p = 1

and

\mathrm{dim\,\,}V=1 d i m     V = 1

is of a special interest. Under this assumptions we characterize lower semicontinuous

1 1

-strongly midquasiconvex functions.