 Advances in Pure Mathematics, 2011, 1, 30-32 doi:10.4236/apm.2011.12008 Published Online March 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Relative Widths of Some Sets of mpl* Weiwei Xiao, Weijun Luan College of Sciences, North China University of Technology, Beijing, China E-mail: wwsunny@163.com Received January 14, 2011; revised January 30, 2011; accepted March 10, 2011 Abstract In this paper, the relative widths of some sets in mpl are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid) of mpl. Keywords: Kolmogorov Widths, Relative Widths 1. Introduction In 1984, V. N. Konovalov in  first proposed the definition of relative widths which is in the sense of Kolmogorov. Let W and V be centrally symmetric sets in a Banach space X. The Kolmogorov n-dimen- sional widths of W relative to V in X (shortly, relative widths) is ,, :=,supinf infnXnnfWLgVLKWVXf g where the infimum is taken over all n-dimensional subspaces nL of X, nN. When =VX the rela- tive widths coincid es with the n-dimensional Kolmogo- rov widths (shortly, nK widths) of W in X, which we denote by ,ndWX. Of course, ,, ,nnKWV XdW X for any set V, and if 12VV, then 12,,,, .nnKWV XKWVX Y. N. Subbotin and S. A. Telyakovskii in [7-9], V. M. Tikhomirov in , V. F. Babenko in [2-4], V. N. Kono- valov in [1,5,6], V. T. Shevaldin in  etc. gained many results in this field. And some Chinese mathematics workers such as Yongping Liu, Lianhong Yang in [15-17] and Weiwei Xiao in [12-14] also did some work on rela- tive widths. Let mpl, 1p, denote space of vect o rs 1=,,mxxx with norm 11=, 10mDD D. Let M be a positive real num- ber, set =:, ,mppMRMDx xx obviously it is ellipsoid in mpl. When =1M, we denote it by p. Theorem A:  For 1p, 1mnN, 1p, 11>nDD, the smallest number M which makes the equalities 1,,= ,=,mmnp pp nppnKMldlD  (1) hold is 101:= 1nDMD, and 10101,0<<,,,= ,.mnp ppnMDMMKMl DMM Theorem 2 For all mN such that >1m, 111 1,, =.2mmKBBl 2. Proof of Theorems Proof of Theorem 1: For 01=,0,,0Dx, we have 01,, supinf inf sup inf inf =1.mnp pppnm nxLlyM LppppyMxpppyM pKMlDM xyxyxy That is 1,,1, 0<1.mnp ppKMlDM M  (2) In order to make the equalities (1) hold, we have that 111,nDDM that is 111nDMD . (3) For 110<1nDMD , we will prove that 1,, 1.mnp ppKMl MD (4) For each =pxDz, 1pz, set 1=,,,0,,0 .nnpyMx MxLM  When =p, the inequality (4) is trivial, so we only need to prove the case of 10M,  1,, ,=.mmnppp nppnKMldlD  (7) From (6) and (7) we get 10,,=, .mnp ppnKMl DMM The proof of Theorem 1 is complete. Proof of Theorem 2: From  we know that 111 1,, .2mmKBBl (8) We want to prove that 111 1,, .2mmKBBl (9) W. W. XIAO ET AL. Copyright © 2011 SciRes. APM 32 By proposition (6) we know that 1111 1,,=,, ,mmmmKBBlK WBl (10) where =0 ,,0,1,0,,0:=1,,, represent the th coordinateiWimii . Set 112:=:= 0.mm mLRxxxx For =0,,0,1,0, ,0iaW, set 111=0, ,0, 12,12,0, ,0,miibLB =1, ,id, when =id, 1i represent the 1st coor- dinate, we get =12ab. So we proved 11,, 12,mmKWBl which means that inequality (9) is valid. The proof of Theorem 2 is complete. 3. References  V. N. 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