On \(\omega \)-strongly quasiconvex and \(\omega \)-strongly quasiconcave sequences (Q657821)

Let \(\omega \geq 0\). A sequence of reals \((f_{k})_{k\in I}\), where \(I\subseteq \mathbb{R}\cap \mathbb{Z}\), is said to be \(\omega \)-strongly quasiconvex if \(f_{k}\leq \max \{f_{k-1},f_{k+1}\}-\omega \) for every \(k\) such that \(k-1,k,k+1\in I\). If \((f_{k})_{k\in I}\) satisfies the opposite inequality instead, then it is said to be \(\omega \)-strongly quasiconcave. Notice that the latter definition does not agree with the one to be expected (\(\omega \)-strong quasiconcavity of \((f_{k})_{k\in I}\) to be equivalent to \(\omega \)-strong quasiconvexity of \((-f_{k})_{k\in I}\)). A sequence \((f_{k})_{k\in I}\) being both \(\omega \)-strongly quasiconvex and \(\omega \)-strongly quasiconcave is said to be \(\omega \)-strongly quasiaffine. The authors give two different characterizations of these notions and introduce corresponding definitions for functions defined on an interval \(P\subseteq \mathbb{R}\), in such a way that a sequence \((f_{k})_{k\in I}\) is \(\omega \)-strongly quasiconvex (\(\omega \)-strongly quasiconcave, \(\omega \)-strongly quasiaffine) if and only if it is the restriction of an \(\omega \)-strongly quasiconvex (\(\omega \)-strongly quasiconcave, \(\omega \)-strongly quasiaffine, respectively) function defined on the convex hull of \(I\).