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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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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:ACCEPTED

14 Aug 2020Submitted to Mathematical Methods in the Applied Sciences
17 Aug 2020Submission Checks Completed
17 Aug 2020Assigned to Editor
24 Aug 2020Reviewer(s) Assigned
02 Jan 2021Review(s) Completed, Editorial Evaluation Pending
04 Jan 2021Editorial Decision: Revise Major
01 Apr 20211st Revision Received
01 Apr 2021Submission Checks Completed
01 Apr 2021Assigned to Editor
05 Apr 2021Reviewer(s) Assigned
04 May 2021Review(s) Completed, Editorial Evaluation Pending
05 May 2021Editorial Decision: Accept