Convexity of generalized proximinal sets in Banach spaces

ABSTRACT

For a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly uniform convexity of are provided for a proximinal set G to be convex.

KEYWORDS:

• Generalized best approximation
• proximinal set
• weakly uniform convexity
• compactly locally uniform convexity

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported, in part, by the Natural Science Foundation of Zhejiang Province (Grant No. LY16A010009). The second author is supported, in part, by the National Natural Science Foundation of China (Grant No. 11571308) and the NSF of Zhejiang Province (Grant No. LY18A010004, LY17A010021). The third author is supported, in part, by the NNSF of China (Grant No. 11101363), Zhejiang Gongshang University (the Youth Talent Foundation QY15-15) and the China Scholarship Council. The fourth author is supported, in part, by Ministry of Science and Technology 105-2115-M-039-002-MY3 (Taiwan).