Characterizations of r-Convex Functions
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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 ∇ 2 f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.
KeywordsStrict r-convexity r-convexity Positive-semidefinite matrices Positive-definite matrices Convex functions Quasiconvex functions
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