We all know the Jensen's inequality is for all probability measures $\mu$ and all convex $h:\mathbb{R}^N\to\mathbb{R}$ it holds that \begin{equation} h\left(\int Xd\mu(X)\right)\leq \int h(X)d\mu(X). \end{equation} But the the definition of laminate is for rank-one convex function $h$, Jensen's inequality holds. I want to find a example that a probability measure for rank-one convex function such that Jensen's inequality doesn't hold.
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What is a rank one convex function? – Deane Dec 17 '22 at 15:37
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The definition of rank-one convex function is: for every $\xi,\eta\in M^{m\times n}$, with $\mathrm{rank}(\xi-\eta)\leq 1$ and every $\lambda\in (0,1)$, $f:M^{m\times n}\to \mathbb{R}$ is called rank-one convex function if satisfying \begin{equation} f(\lambda\xi+(1-\lambda)\eta)\leq \lambda f(\xi)+(1-\lambda)f(\eta). \end{equation} – Min Gao Dec 18 '22 at 13:25