Journal of Shanghai University(Natural Science Edition) ›› 2020, Vol. 26 ›› Issue (5): 834-841.doi: 10.12066/j.issn.1007-2861.2092

• Research Articles • Previous Articles     Next Articles

Extremal representations of polar set and ${M}$-addition

LIU Chang, LENG Gangsong()   

  1. College of Sciences, Shanghai University, Shanghai 200444, China
  • Received:2018-08-29 Online:2020-10-30 Published:2020-01-10
  • Contact: LENG Gangsong E-mail:gleng@staff.shu.edu.cn

Abstract:

The extremal representations of the polar set and M-addition are studied. If K is a compact and convex subset of  Rn, then we have K° = (ext K)°. It is proved that M-sum of K and L is equal to the (bd M)-sum, if K and $L$ are convex bodies containing the origin and M is a convex body in the first quadrant of  R2. Moreover, by the counter-examples given in section 3, all these conditions cannot be removed.

Key words: convex geometry, polar set, $M$-addition, extremal representations

CLC Number: