Higher Dimensional Hermite-Hadamard Inequality for Semiconvex Functions
of Rate $(k_1,k_2,…,k_n)$ on the Co-ordinates and Optimal
Mass Transportation
Abstract
In this paper, we give a new higher dimensional Hermite-Hadamard
inequality for a function
$f:\prod\limits_{i=1}^n[a_i,b_i]\subset\mathbb{R}^n\rightarrow\mathbb{R}$
which is semiconvex of rate $(k_1,k_2,…,k_n)$ on the
co-ordinates. This generalizes some existing results on Hermite-Hadamard
inequalities of S.S. Dragomir. In addition, we explain the
Hermite-Hadamard inequality from the point of view of optimal mass
transportation with cost function
$c(x,y):=f(y-x)+\sum_{i=1}^n\frac{k_i}{2}\big|x_i-y_i\big|^2$,
where
$f(\cdot):\prod\limits_{i=1}^n[a_i,b_i]\rightarrow[0,\infty)$
is semiconvex of rate $(k_1,k_2,…,k_n)$ on the co-ordinates
and $x=(x_1,x_2,…,x_n)$,
$y=(y_1,y_2,…,y_n)\in\prod\limits_{i=1}^n[a_i,b_i]$.
Furthermore, by using the higher dimensional Hermite-Hadamard
inequality, we compare the transport cost in different transport models
on the sphere $\mathbb{S}^2$.
