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ABSTRACT
In a real Banach space X, we introduce for a non-empty set C in X the notion of suns in the sense of Bregman distances and show that C is such a sun if and only if C is convex. Also, we give some necessary and sufficient conditions for a compact set to be the Klee set, extending corresponding results on the Euclidean space.
Acknowledgments
We thank the referee for his/her careful reading and valuable suggestions, which lead us to seek functions such that they and their dual are both Fréchet differential (see Examples 3.15 and 3.16).
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
The first author is supported, in part, by Natural Science Foundation of Zhejiang Province [grant no. LY16A010009]. The second author is supported, in part, by National Natural Science Foundation of China [grant no. 11771397]. The third and fourth authors are supported by Ministry of Science and Technology, Taiwan [grant nos. 106-2115-M-037-001, 105-2221-E-039-009-MY3], respectively, as well as the grant from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Taiwan.