Counter-examples concerning Breckner-convexity (Q6619451)

Assume that \(D\) is a convex subset of a vector space, \(c\in\mathbb{R}\) is positive, and \(\alpha\) is a nonnegative even error function defined on the difference set \(\{x-y\mid x,y\in D\}\). We say that a function \(f\colon D\to\mathbb{R}\) is \((c,\alpha)\)-convex if it satisfies the Breckner-type functional inequality\N\[\Nf\Bigl(\frac{x+y}{2}\Bigr)\le cf(x)+cf(y)+\alpha(x-y).\N\]\NFor \(c=1\) and \(\alpha=0\), the authors construct a function which is locally bounded from above, satisfies this inequality, but fails to be convex. In other words, the statement of the theorem by \textit{F. Bernstein} and \textit{G. Doetsch} [Math. Ann. 76, 514--526 (1915; JFM 45.0627.02)] does not hold in this setting.\N\NHermite-Hadamard-type inequalities for \((c,\alpha)\)-convex functions are established as well, in particular, a lower estimation for \(c>0\) and an upper one for \(0<c<1\). Contrary to the classical case, neither estimation implies the corresponding Breckner-type inequality of the function.