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Higher Dimensional Hermite-Hadamard Inequality for Semiconvex Functions of Rate $(k_1,k_2,…,k_n)$ on the Co-ordinates and Optimal Mass Transportation
  • Ping Chen,
  • Wing-Sum Cheung
Ping Chen
Jiangsu Second Normal University
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Wing-Sum Cheung
University of Hong Kong Department of Mathematics
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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 $\mathbb{S}^2$.

Peer review status:UNDER REVIEW

14 Aug 2020Submitted to Mathematical Methods in the Applied Sciences
17 Aug 2020Assigned to Editor
17 Aug 2020Submission Checks Completed
24 Aug 2020Reviewer(s) Assigned