On a Polygon Equality Problem

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 223, ARTICLE NO. AY O a Polygo Equality Proble L. Elser* Fakultat fur Matheatik, Uiersitat Bielefeld, Postfach 003, 3350 Bielefeld, Geray L. Ha, I. Koltracht, M. Neua, ad M. Zippi Departet of Matheatics, Uiersity of Coecticut, Storrs, Coecticut Subitted by Joh Horath Received October 4, 997 Berius ad Blachard of Bielefeld Uiversity i Geray have cojectured the followig polygo iequality: for ay two sets of vectors x,..., x ad y,..., y i, x x y y x y ij ij i, j i the 2-or ad that, oreover, equality holds i Ž. if ad oly if there exists a perutatio o, 2,..., 4 such that y x, i,...,. That Ž. i Ži. is valid is a cosequece of a iequality that holds i certai Baach spaces ad which was recetly proved by Leard, Togue, ad Westo. We therefore characterize here the case of equality i Ž., actually for vectors i the space X L Ž,., ad subsequetly use this characterizatio to coplete the proof of the BeriusBlachard cojecture cocerig the equality case i a Hilbert space. 998 Acadeic Press *Research supported i part by Soderforschugsbereich 343 Diskrete Strukture i der Matheatik, Uiversitat Bielefeld. E-ail: eua@ath.uco.edu. Peraet address: Istitute of Matheatics, The Hebrew Uiversity, Jerusale 9904, Israel. Participat i Workshop i Liear Aalysis ad Probability, Texas A & M Uiversity, 997. Partially supported by the Ladau Ceter for Research i Matheatical Aalysis, sposored by the Mierva Foudatio Ž Geray X98 $25.00 Copyright 998 by Acadeic Press All rights of reproductio i ay for reserved.

2 68 ELSNER ET AL.. INTRODUCTION I the study of learig i artificial eural etworks the followig easure of welless of represetatio has arise; see Berius : Gie a set of 2 poits i, 4, diide it ito two parts of equal sizes M,..., 4 ad M,..., 4 2 such that M represets M2 as well as possible. As a Ž also coputatioally coeiet. easure of the degree of represetatio, Berius itroduced the fuctio H,,2 i, j ij ij where is the Euclidea or i the real -diesioal space. Berius ad Blachard cojectured that H, 2 0 ad they have also raised the questio, ore iportat to the, of whe H, 2 0. Actually, it has bee kow, by a result of Leard, Togue, ad Westo 3 via a equivalece with a result of Bretagolle, Castelle, ad Krivie 2, that the iequality H, 2 0 holds i a Baach space X for every 2 ad every two sets M ad M2 of vectors if ad oly if X is isoetric to a subspace of L Ž,. for soe easure space Ž,..Itis well kow that the space L 0, p with p 2 is isoetric to a subspace of L 0, Žsee Lidestrauss ad Tzafriri 5 ad the refereces cited therei. ad hece H, 2 0 holds for ay 2 ad every two sets M ad M2 of vectors i the or of L p, p2. Berius ad Blachard forulated the followig: CONJECTURE Ž Polygo Equality Cojecture.. Let x,..., x ad y,..., y be ectors i. The x x y y x y Ž.. i j 2 i j 2 i j 2 ij ij i, j if ad oly if there exists a perutatio of, 2,..., 4 such that yi x Ži., i,...,. Two special cases of the cojecture were proved i : ad ad ad 2. As usual, for ay vector x Ž. T ad for ay positive uber r, we shall let r r ž i / i x r

3 ON A POLYGON EQUALITY PROBLEM 69 ad we shall let l r deote the space of all -vectors x T of real ubers with the or x r. The ai purpose of this paper is to prove the polygo equality cojecture. However, it turs out that the atural settig for studyig the cojecture is the space L Ž,. istead of a Hilbert space. Therefore, before provig the cojecture i Sectio 3 we will cosider, i Sectio 2, the questio of whe equality holds i Ž. i the L -or. 2. THE CASE OF EQUALITY IN L, Let us start with a siple exaple. 2 EXAMPLE 2.. Let X l 4 ad let xi i 2 deote the uit basis of X. Put y Ž,. ad y Ž 0, 0.. The x x 2 y y ad x y for all i, j 2. Hece i j 2 i j 2 2 i, j x y x x y y, but the sets x 4 2 ad y 4 2 are ot idetical. i i i i Hece the BeriusBlachard cojecture is false i L Ž,. ad we would like to deterie exactly whe for the sets M x,..., x 4 ad M y,..., y 4 2 i L equality holds. Our ai tool i studyig the equality case is the followig: LEMMA 2.2. Let s s ad t t be ay two collectios of real ubers. The t s t s s s t t. Ž 2.. i i i i, j ij Proof. We start our iductive arguet with the case 2. We ust show that t s t s t s t s t s t s s s t t 2 2, which is the sae as showig that the iequality s s t t t s t s

4 70 ELSNER ET AL. holds. But this follows fro s s2 ad t t2 by checkig. Let us pass to the iductio step. Suppose that Ž 2.. holds for ad let s s s ad t t t be give. Assue without loss of geerality that t s. The i i i i t s t s s s. It ow follows fro the triagle iequality that i i i i ts ssi sti i i i i i i t s s t s s t t. Ž 2.2. But the 2.2 ad the iductio hypothesis give that ad the proof is doe. s t s s t t i, j ij t s s t i i t s s t i i i j i, j i i i s s t t Ž s s t t. i j i j ij t s t s t s i i i i i i We are ow ready to characterize the sets of vectors x 4 4 i iad y for which equality holds i Ž. i the L -or. i i

5 ON A POLYGON EQUALITY PROBLEM 7 THEOREM 2.3. Let x 4 4 i iad yi i be two collectios of fuctios i L Ž,.. The equality holds i Ž. if ad oly if for alost eery, the uerical sets x Ž.4 ad y Ž.4 are idetical. Proof. i i i i The if part is iediate because i j i, j x Ž. y Ž. i j i j ij x Ž. x Ž. y Ž. y Ž. 0 for alost every Ž with respect to. ad by itegratig over with respect to. Let us the cosider the oly if part. Suppose that equality holds i Ž. ad, for each, let Ž,i. ad Ž,i. be perutatios Ž, 2,...,. for which ad x Ž. x Ž. Ž,. Ž,. y Ž. y Ž.. Ž,. Ž,. We will eed the icreasig order of these ubers i order to effectively use Lea 2.2. We ust prove that for every i, xž, i. Ž. yž, i. Ž. a.e. Ž 2.3. If Ž 2.3. is false, the there exist 0, 0, ad a subset with Ž. such that Ž, i. Ž, i. i for every, x Ž. y Ž.. Ž 2.4. Note that, for every, the expressios x i Ž. y j Ž. ad i, j i j i j ij Ž x Ž. x Ž. y Ž. y Ž.. are ivariat uder perutatios of x 4 ad y 4. Therefore Ž 2.4. i i i i, Lea 2.2, ad the equality x y x x y y 0 i, j ij

6 72 ELSNER ET AL. yield the iequality 0 x y x x y y i, j ij ½ H i j i, j x Ž. y Ž. ½ Ž i j i j. 5 x Ž. x Ž. y Ž. y Ž. d ij H Ž, i. Ž, j. i, j x Ž. y Ž. Ž Ž, i. Ž, j. ij x Ž. x Ž. H Ž, i. Ž, i. i y Ž. y Ž.. d Ž, i. Ž, j. 5 x Ž. y Ž. dž., a cotradictio, ad our proof is doe. Reark 2.4. Suppose that the easure space Ž,. is a copact etric space ad is a Borel easure. Assue also that x 4 i i ad y4 i i are cotiuous fuctios. The the proof of Theore 2.3 shows that the equality x y x x y y 0 i, j ij iplies that x 4 4 i i ad yi i are idetical for every. Lea 2.2 ow yields the followig result for ootoically icreasig sequeces of fuctios: COROLLARY 2.5. Let x 4 y 4 L Ž,. i i i i ad assue that x Ž. x Ž. ad y Ž. y Ž. a.e. The i i i j i j i j i i, j ij x y x y x x y y.

7 ON A POLYGON EQUALITY PROBLEM PROOF OF THE POLYGON EQUALITY CONJECTURE Let us ow cosider the polygo equality cojecture. Suppose that Xl ad x 4 y 4 X are such that Ž.. holds. Let 2 i i i i 4 S ux u. Furtherore, let deote the oralized rotatio-ivariat easure o Ži.e., Ž... The the itegral H ², y: dž y. depeds oly o ad ot o itself. Put The the ap defied by H c ² u, y: dž y., u. T : l2 LŽ,., Ž Tx.Ž. c ² x, :, is a isoetric ebeddig of l ito L Ž,. Ž 2 see Lidestrauss ad Pelczyski 4, p. 32, Prop Moreover, is a copact etric space ad, for every x l2, Tx is a cotiuous fuctio. It follows by Theore 2.3 ad Reark 2.4 that, for every, the uerical sets Tx 4 i i ad Ty 4 i i are idetical for every S. Cotiuig, sice the expressio x y x x y y i, j ij does ot deped o the orderig of the x i s ad y i s we ay assue that both sequeces are arraged by decreasig order i the ors, viz., x x ad y y. Suppose ow that y x ad let y ys. As etioed earlier, by Theore 2.3 ad Reark 2.4, the sets Tx 4 4 i i ad Tyi i are idetical, ad so for soe j, we ust have that c ² x, : Ž Tx.Ž. Ž Ty.Ž. j j ² : c y y, y c y.

8 74 ELSNER ET AL. It follows that ² : ² : y x y x, y y y, y y. j j This iplies that x y. Clearly, we ca assue without loss of geerality j that j. Sice j j j j j2 j2 j2 j2 we obtai that x y y x y y x x, i j i j i j i, j2 2ij x y x x y y. Repeatig the above procedure, we get that there is a j, 2j, such that x y. Cotiuig i this aer, we arrive at the equality j 2 i j i, j x y x x y y, which iplies, by the triagle iequality, that either x y ad x y or x y ad x y. We have therefore proved the polygo equality cojecture. We coclude with the followig coet. For ay two sets of vectors 4 4 M x,..., x ad M y,..., y i l, defie the fuctio i, j ij f M, M x y x x y y. Fro the foregoig we kow that fž M, M. 0 ad that fž M, M if ad oly if there exists a perutatio o, 2,..., 4 such that yi x Ži., i,...,. Oe ight therefore be tepted to cojecture that f, defies a etric o the subsets of cardiality of l 2. That this is ot the case is show i the followig exaple i which trasitiveess fails to hold. Let ž / ž / ž / 5 ½ M 5, 5, 5, ½ž 0 / ž 0 / ž 0 / 5 M2 3, 3, 0, 0 0 0

9 ad A coputatio ow shows that ON A POLYGON EQUALITY PROBLEM 75 ž / ž / ž / 5 ½ M3 2, 2, fž M, M2. fž M, M3. fž M 3, M REFERENCES. H. C. Berius, Traiig ud Geeralisierugsverhalte Neuroaler Netzwerke, Diploarbeit, Fakultat fur Physik, Uiversitat Bielefeld, J. Bretagolle, D. Dacuha Castelle, ad J. Krivie, Foctios de type postif sur les espaces p L, C. R. Acad. Sci. Paris Ser. I. Math. 26 Ž 965., C. J. Leard, A. M. Togue, ad A. Westo, Geeralized roudedess ad egative type, Michiga J. Math. 44 Ž 997., J. Lidestrauss ad A. Pelczyski, Absolutely suig operators i Lp-spaces ad their applicatios, Studia Math. 29 Ž 968., J. Lidestrauss ad L. Tzafriri, Classical Baach Spaces II, Spriger-Verlag, New York, 979.