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Hausdorff dimension of the nondifferentiability set of a convex function

Mirzaie Reza (Department of Mathematics, Faculty of Science, Imam Khomeini International University (IKIU), Qazvin, Iran)

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ≥ 2, has Hausdorff dimension at most n - 2.

Keywords: Hausdorff dimension, convex function, Riemannian manifold