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arXiv · 1512.05529

Epigraph of Operator Functions

Mohsen Kian
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Abstract

It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in\mathbb{R}\times\mathbb{R};\ f(x)\leq y\},$$ the epigraph of $f$ is a convex set in $\mathbb{R}^2$. We state an extension of this result for operator convex functions and $C^*$-convex sets as well as operator log-convex functions and $C^*$-log-convex sets. Moreover, the $C^*$-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.

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Mohsen Kian. 2015-12-17. Epigraph of Operator Functions. https://arxiv.org/abs/1512.05529

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