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Characterizations of \(r\)-convex functions. (English) Zbl 1231.90314
Summary: This paper discusses some properties of \(r\)-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of \(r\)-convexity is given without assuming differentiability. The concept of strict \(r\)-convexity is introduced. For a twice continuously differentiable function \(f\), it is shown that the strict \(r\)-convexity of \(f\) is equivalent to a certain condition on \(\nabla ^{2} f\). Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.

MSC:
90C25 Convex programming
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