A note on entropy optimization (Q2759574)

This note is devoted to an optimization problem in a nondecomposable Köthe space \(L_\mu(\Gamma)\) of real valued measurable functions. Given an appropriate lower semicontinuous convex function \(h\) and the integral functional \(H(u)= \int h(u(s)) d\mu(s)\) on \(L_\mu(\Gamma)\), the conjugate \(H^*\) is performed by \(H^*(u^*)= \int h^*(u^*(s)) d\mu(s)\). This result combined with Fenchel duality is applied to obtain an explicit form of the optimal value of the problem NEWLINE\[NEWLINE\text{Inf}\left\{H(u)- g(\mathbb{V} u): \int_{\mathbb{R}^n} u(x) d\mu(x)= 1\right\}NEWLINE\]NEWLINE when \(\mu\) is the Lebesgue measure on \(\mathbb{R}^n\), NEWLINE\[NEWLINE\mathbb{V} u= \int_{\mathbb{R}^n} (x- x_0)(x- x_0)^t u(x) d\mu(x),NEWLINE\]NEWLINE and \(g\) is some concave function.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00036].