View of Duality relations and characterizations of best approximation for p-convex sets

(5)

^iro

: _E {-r,'-" (l ₎

Norv, using

the relation, t+l ^"a-t5),

^rve

can

easily

^

Le1,

us

supposc

that /(rr) :

_e,(n)

: nr¡

ø e _10, 11.

:0, 1, ...¡lL. In this

caso, Trorn (5)

ii,

results

that

(6) ^% : ^t. I ,-1)i-'/lnl"".

It'v:O \U/

Using

the idcntity

(see

[7],

Problem 189,

p.

42)

É t-ti'

)t

g4 DORIN ANDRICA 2

_

^Since

^the

^operators

I

ancl

Â

comrnute,

applying the binonial for- rnula, flom (3)

rvc get

(4) t),(f

_{; n)}

:

,\^(r)

^{L, Ío *,.}

lYe recall that

(sec

f3l ^pp.

^3a)

IL\TIlIl\IrVI'I0¡\ - IìE\¡tjll l)''\N¿\LYSD N[.rIIIilìIQI]lj lì'r ^I )E ^.l.l ltio n Il,: I)E r.,,{pptìo-\ ItrA'l' IoN

tr,,ÀN¿LLYSIt

NUtuiìRIQUE rì'l' LA 'l'Hit0ntH lllì ^L'ÂPt

-flOXI[,IATtrON 'llorrro

16, N' 2,

1gl}7, ¡rp.

Ð5-lll8

DU-A.I,ITY IIE LÄT

IO N S

ÄND

^C IÌ-A Iì AC

llE

R

IZÄT

IC

NS

Or' BDST API'ROXIMATION IiOR ^{p-CONVEX SIITS}

S'|IìF,A.N COIIZASj and IOAN ¡{tlNTlìAN

(Cluj-Nrpoca)

1. Introductiol. A

^subset

of a vector

spacc

is said to be

^conuefr

if

together

u'ith any

1,rvo of

its

points

it

conl,ains

the

rdrolc

line

segrnent .,ìoining

them. J. von

Neurnann [16],

in

conncction u,il,h t]re

inttoduction

,oT

locally

convex topologies,

rcquiretl that onl¡'thc

rniclpoinl,

of this

^seg-

ment

belongs

to the given

set,

tlefining "2

so thcr

a -.otl."* sots. J. \'\'.

'Gleen

antl \\'. Gustin _[8-l tlefinctl and studictl thc

cluasiconyex sets ^: a, sel,

in

a

vcctor

^space

is

called quasicon,aen

if

tclgcther

rvith an)' tu'o

of

îts points

,r ancl ¹⁷

it

contains

all the

points. tlividing^

lhc

^segment

ln, yl into

a

ratio

belonging

to

a prescribecl set

A

^c'10,

1[.

Sotne extensions of the

notion

of c¡uasiconvcxil,y rvere gir.en

b)'4. r\loman

_[1'l anrl Gh. loacler

1221.

flhis

^paper

is

concernecl with ?-corìyox ^sets,

i.o.,

cluasi-convex ^sets

rvitlr A : _I'p),

n'here

p is

a given n rrnbel

in

_10,1

_[. fn

othor tt'ords,

a

set

Y in

a

vector

^sp¿rcc

is

said

to

be p-cortaen

il py ₊

⁽⁷

' ^{p) Y} c Y.

^.I.he

t.-"or.-"*

sets

or

rniclconvcx

or also centretl-convex

[1

?])

^rverc recent- 2

l¡'

^userl

ⁱⁿ the study of the continuity of

Jensen convex

functions ([5], l21l) ^{and of} the

stabilit-v

of

Jenstn inequalit5.

([3]).

Ono

of the

authors c¡f

tìlis

pâper provecl

in _f1a]

some soparation and

support

theorems

for

tr-coìlvex ^sets

in

topological r.t¿ctot'sp¿ùces,

antl

gavc

in _{f15l a}

^LagrzìIìgo

rnuitiplicr lule

^il-L

_?-convex

programming.

ì{. P. I{ornelcltuk _{[12], pp.} 28-33,

proved bhe

follorving

^('cluality

rolal,ions" for the

best approximal,ion b-v elernrnts

of

convex sets:,

îr'rnonnrlr

I.7. IT Y is

ct, ttonaoid co'nueo sttbset, of a rea,l. nornr,ed space

X,

^tlt,en,

a)

Ú'or euet'y ,r

e,Y,

^th,e

tluality

^t'elal,ion

inf

_{llø

- yll:

^{y e}

y}:

_¡iuP

_{rt(ø)

- supl*Qt),y ey'r: øt'e

-ll*1,

Jt,olcls,rult,ere

ß*: _{r* eX*; llø*ll< t¡ ts

ttte ck¡seil, u,ttit

ba,ll itt

tlte ^cluct[, sp&ce 'Y* ^o.f 'Y ¹ antl

compute B,(e,;

n)..

I'lrcn /, :

^{vs/'rs, v}

:

(7) ^L

1

i-1)":i!8;,

vhere

^¡Sì arc

Stirling's

nurnbers

of the

seconcl

kind, florn

(rt),

(5)

ancl (6),

rvc

obtain

tr,,(e,,,) _ É (i) _{;j ",,,}

]ìIìIì]iRENC]]S

[1] ^{r\ tt d r i c} a, 7)., Poruers lty l'lenrsteitr's opetolot's utttl sontc contbinatot ial 1:ro¡tcrtics, lLÍte- tant sentitlar ou Jnnc[iona] cc¡rratiorrs, approxinral-ion and convexity, Cììuj-Napoca, Plc- print No. 6 (1985), pp. 5-9.

[2] ^{(l lt an} 9,, G.-2., l]crttslcin polyrtotnials uia tlrc shìftitr¡1 opclalot'. Thc Amclican llathcr¡n- tical i\,Ionl-hly, $f ^(1984), rrr. 10, 634- 6iìåì.

[3] ^{G Ìr c I I o tr} d, ,{., O., Cnlculttl ctt difcretrle fittite , Iidif,.'fcluicir, ì3uculcçti, 1956,

[4] II c, ^Il ., ^{'1'lte powcrs} tlttl tlteír lltrnslcitt ¡tr.tLyttotninls, lìcal Anaìysis ExcÌralgc, tl (1983- 1984), no. 2, 578-58:1.

[5] l{arlirr, S., Ziag,ltt',2., llt:tttliottol'l>ositiucA¡tprot,intoliottO¡tcralrtrs,.I.r\¡rprox, 'flrcor'¡., :l ^(1970), :ì10-:ì39.

L o ¡ c rr l?., O., (',., llcrnsleitt Polytt\tnials,l-ìniÌcrsily of f'oronto Prcss, 'l-oronto ^1956,, P o ly a, Cì., S z c g o, Cì., Problen¡,s atrl 'l'ltcotc¡tts ítt Attctlgsis, \'ol. I, SptjnS-cr-\'cllag"

13erlil, -Hciclelbel'g, Nctr' \or'ìr, 1972.

Iìcceivcd 20. III. 1986

U ttiuctsilalca ditt CIuj-\'ctpoca IÌ acu I la lca d c nt aletnal i cít

S lr. l{ogidrticclntL, 1,

3400 Clu.j-Napoca llontd¡tict

(ì 7

96 ÐT. COBZAS ^aÌ1al r. tviuN.TEaN 2

_

]1)

for

0Ì:t2t"!l ;))

e-f\Y,'tl¡ure

ett¡,ists ar¿rl

,t:l in X* uittt

llæf l]

:

1

suclt, tlta,t

. ^inf

ìll¿

- ^yll:y eY):,r;*(,ir) -sup

^[a;g'(y)

^{:y eY).}

In

Section

2 of this

ptìpet n'e shall

shol'that

llheorern 1.1 lcrnai¡s,

true if thc

^sc¡1,

Ï is

supposrxl

to

lto onl¡' p-conr.ex.

Section 3

contains c¡x-

tensions

to

the _?r-couvex

Ligun

ancl

\r. G.

I)oronin conica,l constraints. Sectio oli 1u-caverns (subsets

of

a

?r-convex scl,

\'ith nonvoirl

inl,er,ior')

extcnding

sonto resull,s

provc¿ by

C.

}'r'anchetti

and

l.

Singer

[ti] in ^tho

^{conr.cx case.}

\\¡o

shall nct¡rl the fclllorving

three kllon'n

I'csults ^:

lùruor¡nrl

1.2

([]4'1, Thcor,un 4.4)

. I.f y,is

ct,tt,tntooitl p-cotùDefr sul¡set o.f ct, real, loctr,ll,y co,n,,De!r) spcu:e -Y ^{u,nd, ít:o e}

-\:¡Í,

^{then, t,het,e eerisls} ¿x e

-y*

srtcl¡

llt.al,

irrf

_i,i'*(y) :

y e Yl )

,rr*(,r:u).

lnnorr'nrr 1.3

^(114;1,

llheolcln 1,2). r.f 'r is

^a, J)-cotùI)eü ^subset, uitlt,

't.¿ot¿.'ooitl

i.nlerior.rI

u, retil, locc,lly ft)lLr\c,tt spuce, ^lher_L e'uery ltrtrrrlury poí,nt ç.f

I is

utn.tuittetl.

in

^tt cktsul ltyTterplt,ntt su,.pltorlitto ^'T .

'llnnol¿r,;ilr 1.4 ⁽ _fl.'1, fllheorun

J.:j). r.t jl is

a ?-cotuLr0!Ð sultset o.f

a

tolto- loç1icøl t:ectot' s1ta,ce, tlten,

a)

th,e closut'e.

Y oJ 'Y ís

cctnuen;

. ¿/).i.f_tte\l,1leint lf antlO {ø{ Irtltclt d.ìt)+(l-

d")y e

inty;

in pnrticrllar,

_tltc itttcric¡r _e.f ^'Y

is

convcü.

All tìre t'cctot'

spâces consitlered

in this

^p¿ùprrr

rvill _{l¡e taken}

oyer

thc fielrl

7ù

of leal

nurnbct,ls.

2: Dualit¡'

relatiorts I'or iu-t'.ottvt

x

srts.

l\/e

^ncretl i,he follou'ing _exten-

sion of a l,ell-lrnot\¡rì

separ'¿ìt,ion l,heolern

fol

^p-cont'ex sets :

'llrlnonRx 2.1.

Let

^'Y

untl Z be tu¡o

notll)oicl antt

disjuùtt

sntbsets r¡.f

a-toltolo¡1ical_aectot'space,Y.

I"f

^'Y

is

p-con,l)et) ancl

z is

coiu;en

(,nil

_open.

tlt,en, Y eutil

Z

cu,nbc sepura,ted hy a ctolctl lt,y7ter.1tlu,nein ,y.

I1'oo,f

. R¡'îh<xllenr

1,4, ø)

, ^the

^closur,e

Y of Y is

convex.'\4'c statc

that Yî Z:

Ø. Intìe,crl,

!f

^e;e_,l.

n Zrt)ten,sinceZ

isaneighbour,hoocl

of u,,it lt¡ulrl

^.l'ollol'

Lhnt-Z ì y +

Ø, rvhich

contratlicts thdltypothesis of thc

l;heolem.

Nol., applying a

clir,ssical sepalation _theolc¡m

(tbl,

Theo- 1er1 .9-.1),

thc

sets

Í

anri-z'aaìt'be sepa,ratt,cl

ïy

a closerl tryperiìianô

in ,r-

It

follo.vr.s

tliat ^Y

xnrl

Z

ale_,scrparatt.d

too by ltris

hypcrplâire.^l

.I.ho nt¡x1, lerur--ra

is

rvoll ktrou.n ancl eas.v

to plove

(seci [12],

p.

B0) :

Irlìttìr¡l

2.2.

I.f I is u

t.to¡'tnetl s,ltücte,

rl}

^cvntl

n* eX*,

l,lten

sup _{ø*(,ø) | ^ln

€,Y, _lløll < r) : r.

liø*ll.

Non', we statc thc fit'st r.luality

theor,ern ^:

fllrtr,lon'llrr 2.3.r1-

Y is

a tùorL1)oial p-cotlDex) s,ubset qf anortneil space

x

and

n ^e

X,

tltetr, th,e .follotui,ttg d,ttu,lí,ty relq,lion l¿okl,s:

(1) ^infiilø - ^yll:Ue y):sup

_{:u*(a)

-sìrp ^{u,r(ù:ye Y):n*eI)*},

I,-CONVD>( SÁiTs 97

coh'ere, P;*

:

_{.r* e

-{* :

fja'F

ll <

^I

} ^is

^l,Ì¿e alose<l u,t¿i,t, l¡all

in

th,e durel sptace

Xa

o.f

X.

ProoJ.

Pttt

i:ilrf{ll.r -all;ye Y}, I:sup[,r*(;rN -sup

^[e'o(y)

:yeY):tx*e]]x].

Since.

¡¿,'r(.r)

- ^sup

^l,r{r7)

^{ty e} Yl (

æ*(r)

- ^ns(y') (

_ll,r;

- y'll fol ^all

&*eIl* and all y'eJl , ^{\\'o ltar.e s} (i. _{¡\s 0e -/l*,} it fó[orvs

thal,

s) 0,hencei:sinthecase i:0.

Suppose

nsrv thati>0. ^{l3y lheo-}

tem 2.1,

the

¡"r-convex

set Y

ancl

thc

open

ball Z :

_iz e

-\t: llr - ^{zll <i1\}

can

be

separntetl

b¡' a

closotì h¡'pel'¡lla,:re.

lhercfore, thcrc

cxist, a,ik e

.r*

with

_llø+ll

:.l

anrl c e lJ

slch thal

(2)

;irff(.q)

g

cr

q

,zro*(e)

tor all

^y

e l!

ancl

all

^p

eZ.

tsy (2)

Íùn([ l./eryìrnn 2.2, n'o ollta,ilr

sup{rff(y) :

y e !l} (

inf{*;fi(z) : ø e

Zl :

infl,ró*(,¿

-

^{to) t}

^{u e,y,}

jl¿oll

< i):lt::lÇt;) -sup{*;f(zu) :tue-,[,

_llzoil

<il :,rf(al -

i.ll¡rfflf,

which irnplit's

thnl,

i -- i'

lløffll

<

,r.f(,r)

-

sup i,rl'(y) ^{: u}

e Y) (

s.

illte

itreclnalil,ies

s ( _{rl and} i { s

g-ive

i - ^s

rùrì.(t'r'ht¡orenr 2.B is

Ilro\/en, I

'lhe.

Ïolktrving,tlrlorern

contains

a

conclition

ensuring

tha,t

the

su- prcrìr.unr

in thc rislrt-han(l of lclation (1) is

¿r,tta,intrd:

-

'J'ttEor¡lill 2.4.

A¿diile to llte

ltypotheses oJ ?h,corent,

z.J

th,e conditiott,

n.Él't

therc e¡¡:tists,rf; e -T*'zriritlr, llrf

ll': I

^{su,chi that llte sccotxl} tlua,lit,y rela- l,'ion lrckls :

(3) inf {ll,i;-yll

^:

ye Y):,rf(ø) -srp{¿ü(y):yey}

l?roo.f . r(ggpinu l,he notat,ion

in the

_¡rroof of Theolern 2.8, l,helc ,rxists

¿ù $cquence (¿:;l)"e N

in Ii*

such

that

(4) lirn

_[a,,](,u)

--

_{sulr {c;T(y) :}

_{y e}

_{y,r1 =: s.}

Iì.y

l,Ìre ,Alaoglu-lìoulbalci t,l.reorc.nr,

the

closerì

unil; ball /J* is

¿ü*-corìr-

pact,

so 1,h¿¡1,

there

exisô

a

subncl, (.r;)*)rer

(/l is a

rlilt.ctc<l

set)

oli

tlic'

sequerìcrì (,r;f ) ^a,nrl an elcrlrrrnl; ;rf;

of

,IJ* such

t

ra,l,

(5) lirn *',fr(r') :

.r:Í(*r')

for'

¿r,ll 1t,

e,y.

I'r'onr (4) antl (5) rvith

¡6t

:

_{;y, u,e dt:rive}

_thnt

(6) lirn

sup _{n*

^(.!/')

:

y,

e ^y _rr

:

øf(*N

_

_s.

3

u8 FT. ^{COBZAS arìcl} ,I. _ÌVIUNTEAN

Ry

(6) anrl

(5)

rvit,h

r' -

^,!/

^€ ï, it

^tollorvs

that

ri@) -

liur.,¿,1¡(J) l:eIi "

( lirnsup

_{,rté 1i}

l.r,f^(i7'):U'e

^Y.|

: øf(ø) -

^s.

lhcrctore, sup

_[rr$(ø) :

y

e

Y,ì ç

^;rrf(e;)

-

s or', ecluivalently,

(7) s(a,ff(r) -stì.p[nt!):yeYt¡.

'Jìnking

into

account

the tleïinition of

s,

relation

(3)

is

a consecìucnco

of (7) antl

Theorern 2.3.

1o finish the ploof,

$'e hâ\'e-. 1,o

shou' that

_{lløf Il}

:1. Sinco

ø t'

)',

i1,

follor.s

th¿r,t

s

^---

i> 0

ancl,

by (3),

e;{

+ tl.

Supposing tha,l,

llrf

ll

<

¹

{n'c ^lcnotr' l'hat

^e;f; e _-11*,

i.er.,

_ll¿'f

ll < 1), x.e ha,ve À'erf e Il*,

^u.ltore

).

:

_{lla;f ll-1}

> 1,

anrt

s

) (À';¿f)(¿) -

^sup

^{(i''

atff)(:ry)

:y eY,\: l's}s,

thich is

a contlatlicl,ion. 'I.he proof of Thc'or-cm 2..t

is

complete. þ

If Y is

a sul¡set of

a

nolrnc<l spzr,cc

,li and

lr ^e -\1,

l;ht'n a

pro,iection

of ¿ onto

Y

(or a

best u,1tytron'intation elcn¿ent

of

^,r

in ^T) is an

cleurent

lt ey

snoh

th¿rt ll,t,-- ltll <

llø

- ^{y'll for all} y'eY. Frorn ilheotem

2.4, one can

delive

a chalactelizntír:rt of

plojcctions

onto p-cotrr.ex scts ^:

Oonolr,.tr¿v 2.5.In,ortlertl¿at

ylte

ø'pro.jectíon oJ

r onto'Y it is

suJJ'i-

cietú, ctnrl

i,f Y is

'p-cotrlt)e.t:, ul,so ttccessnr'y to eæist e;if e

,¡'r

ui,tlt, th,e p?'opcI'-

ties; r¿)

_{lle;f ll}

- 1; ^b)

^pf(ar

- ^y) -

^ll,r,

-'yll; c)

^erü(y)

:

sup

i,rf(y')

^:

!'

^e

^yl.

Proo.f

.

Suppose

that rj'e -f* verifics

condilions ø), i¡) arrd,c).

lhen

1ln

- ^yll :'rt@ - ^y) - ^nt;l - ^y') + [¡,f(y') -

^¿-,*(y)]

^(,r;f(r - ^y')

^<

<llrf _ll

_lln

-!J'

^li

=

^ll,r

-

^!J'l[

^{for all} y'in'Y,

x'hich

shou.s

that _¡l is a

proje,ction

of

r¡:

onto

Y.

[)onverscl¡-, supposo 1,ha,t

Y is

p-conr.ex

antl lct rl be a plojcction of

ø orLto

Y.

_'Ihe

functional

øf;

in

Theorern2.4 verifies

that _|lrf

_ll

:1

anct

(B) llr'-

^y ll

:

ø¡*(r) -= sup

{¡ü(y') :y'eYl.

It lernains to shou'that

arf(y)

:

sup

i¿f(y') :

^'!J'e

Y].

Oi,hclrvisc, lud'(y)

<

^snp {er¡*(y') :

y' e Yl, r.e

^hâ,r'e

ilø - ^yll : _{llrf ll llr} - yll>ni@ -a\ -

^øf(a;)

- ^n't(y)>

>

øð'(¡)

-

^{sup [e,f}

^(y')

^:

^{y' e Y],}

rvhich

contlzl,dicts

rclation (8). I

Florn

Clcllollaly 2.5, one ciùn

obtain a tcll-huox'n

charactcrrization

of ploìcction onto

corì\rex subsets

of Hilbelt

spaces :

r+

Conolr,Àlìrr

2.6. I'et ,Y

be

u ^Ílilbert

space,

Y. f,

ø e

-T\)l

^ørzcZ

y e y. In

ot"cler tlt.u,t

y

be ø pro.iec,tiott, o.f

n

o'ttto

'I, it is

su,,fficient ctnd,,

if Y

is pt-conuer, also tlecess(r,ry th,ut

(u - yly' - y) ^<

^{0 .for}

^all y' in

^Y.

p-CONVEX SDTS 99

Proo.f.

SuJ.ficiency.

We

have

llu -

^U'il'

: (r - ^y'ln - ^!t') : (a -y ^+y - ^Y'ln - ^{y + u} -y'):

:

_lln

- ^{yll" + 2(n} - ^yly -

^U')

⁺

^l[y

- ^{y'l[' Þ}

^ll,u

-

^Yll'

for all y'br

Y, n'hich shorvs i;hai, y

is

a projection

of

a:

onto

^Y.

Necessí'ty.

If y is a plojection of

^er

onto Y, thel'c cxists rf

^e -tr*

verifying conclitions

a,),

b) and c)in Colollary

2.5. 13¡' Rie'sz's r'(rpresen-

tal,ioir

t'úeorenr, there

is

a

uin,ll

^such.

that

ll'¿¿ll

: _llrf

_ll

:

1

antl ntþ) : :(zltr,) for all

øe

-Y. then (n-yl'rr,) :øf1- -tt):llæ,

^All

: _llæ- - yllllu,ll,

n'hich shou's

that jn the

Schrvarz inec¡uality onc has

the

equa- ø e _ives

_y)),, Jì

suclt

_(y it ^follorvs tlrat u : that

^q.('t:

-

^o

^y). :

^Since_lln

- - ^lt'ltt) )

0

fol

all t¡'

in

^Y,

0, n'hiclr irnplies

that (r - ^yly' -

Directly (i.c., rvithout

appcaling

to

^Coroilary

^2'õ)' one

carì prove

a slightly nìorc

genct'al result :

'

lùrìoposrlrroN 2.7.

f'd

,Y be

a pre-IIitltert sptlcc, Y

^c-

X,

^{ø e}

X\Y,

a,tlcl t1

e Y. In,

oriler tltat

t¡

he ø proj'ctiort, o"f t:; ctrtto

'I _{it is}

su/ficiey,t,und,

i.f

Y'

is

'p-cotl't)e$, tt'l,so tt'ecessq'r1¡ th'at ^(:t¡

-

^y

^ly' -

^'y)

(

0 .for

all y' in ^I

^.

Proof. _'I'he proof of

the

sufficiency

part

is the sarne as

fol

Colollar]:

ttecessit¡',

r'elnãrì< t)taL 'p"y'

+

⁽¹

-

^pto)

^y"

^e

^Y

and all n,

^Llle

^{property is true for}

^r¿

:

1

onvex. As

P^Y'

I ⁽⁷ -

P'')'Y" e

Y for

^{¿r, /c}

t,lral;

p"* nr)!/" _p(pr!t' +

⁽¹

-

pr)!J")

i +(1 - p)y"e Y. illhcrefote p"!l' +(1 -

1t"\'!1"

e Y for all'r¿e

^-rY.

¡o11',

if

^q is a

plojection

of ø

onto

^Y and _lJ ^e'X, then

p"!l' + (I -

^{p")lJ e}

e Y irnplies that

ll,¡,

- ^{yllt <}

^ll¿,

- ^p"!t' - (t - ^p,)!tll' -

^Iln,

- ^lt - p"(!t' - ^y)ll' : :

_ll,r

- yll -

21t"(;a

-'Ulll' - ^!l) I Jt'''llll' -

^Ull'.

thcrcfort-¡r -

2p"(:t:

- ^llly' - ^{y) +} p?"ll!l' - yll'Ì) ^{0 or'} -

²⁽ⁿ

- ^ylíl' - - ^{y) +} p"ll!l' -llll'>- ^{0, 'Jhhing}

^7¡

⁺

^{@1 one}

^obtains (r-:/lll'-y) <0. I

'l'htr next

ex¿¡rnple shon's

tìrat the p-colrrexity of I/ is

cssential

fol tho valiilit)'

oli

the

neoessit-v

pali, of Corollaly

2.5.

ììxÄrn,t,o 2.8. Ìret -L : Ilr y: [-1, ^7], t : 0¡ ll : l.

'I'lten

inf{lø - ^!t'l

^|

^4' e'Il :

_l,rr

- Ul: L ^{suppose thele exists}

^{aff eHa'}

: : .Il veliäying

corrriitions

ø),

Ò)

antl

c)

in Colollaly 2.õ. then

_I;rf |

: I

and 1 :

^t$

- ^yl : r,åt(-1), giving the

cotrtladiction

-1.1 : øf(1) : sup[( 7)!t', y'e.ll]:1'

lìr,;u,tn,r 2.9.

In the

case

of the

sonl'ex se1,

Y,

tl'heoretns

2.3

ancl 2.4 rvere proved

by N.P. I(orneichuk

_[12],

pp.28-33,

ancl Corollary 2.i-r

ty G.Sh. Rubinshtein fl8l

^{(see also}

^A.

I,,. Garìcavi [7]).

4

1 (10 ST. COBZAS ârid I. MUNTIEAN jp-col.lvlcx sE]s ₁₀₁

Let

,Y be a norrnecl space anrl

let 1l

btr

a

^c.ou(ì

in _-.f. ^(ìivetr

n, suLt- space

Z of X,

denote

by Z'

the algebra,ic rlual of

Z.lto¡'.¿'

i¡t

Z' s'c

prLt

(11) _lllø'lll :

sup{ø'(a)

: ^?€Z

ⁿ

(-i'') arrtl li"ll <

¹¹

(the

case

lll¡' lll :

oo is

not

exclu<Ie:rl).

It

is eirsil,.i.' secn

th¿t if

ill

r'

_iil

< _"o,

thcn

(12) ¡'(ø)

<

lllø'ill llall fol

a,ll

s in

Z

n(-Ií).

Also,

if

^B'F e 7t*¡

i.c.,

z'F

is a

<tolrtinuous lin,.-ril,r,frrnctioltal an

Z,

t]¡,.,ttr

(13) _lllp*lll {

llø'kll, rvhcre ilønll ⁼⁼ sup{lr'r'(r) t,:,; _e

Z,llzll <

^11.

Di:note also

(11) u; :

_{z'

eZ' : liie'l ^<

^11.

Nolv, u'e

al't¡

in ^position to state tlrtr nrailr lcsult of

l,hjs secl,ion:

lltn:oI¿lr,u

3.3. I'et

.Y lte. ct, r¿rtrnt,etl s1tooa,, let,

I( hc

^((, (()n,,ï(,l) con,e

in,

X,

^l,et

Z

ltc ct subspace o.f ,Y

and

l,et

Y

be

(,

p-c()tt,t)e,1; sttbsel o.[

Z

^st,r,c:lt,

'that (u I I() n ^{Y +} ø _.for øll

^,t;

eZ. I.f

Lh,c d,istctnce .f,un,ct,ion,

tl6(,,Y\

i,s cr¡tttitnlous at al,east on,e poin,titr,

Z

rel,cr,trí.uel,y !,o

Z,

^{tlt,en, tl¿e} du,s,Iitlt relotí,ot¿

(15) tl.¡(r, T) -

suplz;'(r)

-

stip{z'(ø)

:u

e

Yl

:

z'

^e

}}),\

It,okls

lor

^{¿t,Il, tt iu,}

/'. I.[,

]n,()reou(ìr, ,l; e

Z \.

^ll

,

then there e,t:isl.s zf; e Z¿'tuitlt, llieii'ill

: l,

^stt,clt, lltot, tlt,e.l'irst stt,¡trernunt ín, tlte

riqlt.t

sitle o.f

(i5) is

q,chie"^etl

ut ril,

i.e.,

c|¡¡(:t:,

T) -:f(er) -supfziF(u): ue)ll.

]"roo.f.

Fol

^rr

eZ,ltrrL

/J(lr)

: _fl¡(:u, Ï)

a1rl

(16)

r9(;r)

:

_sup{ø'(ø)

-

suptz'('ll)

t _lt €y) r â'c 1l;) : -

^sup[inf

^lz'(c - ^{y): lt e Y]}

^{: ø'}

^e

^Il¿1,.

Iìirst,

rve shall shorv 1,ha1,

(17)

r9(ø)

ç

^ll(e;).

l3v (9), Iì(,n)-inf{ll _-/,;ll : lt,e7(, irf

tue

Y}, ancl t¡-n+it_-- -fieZ lor ali

^{À: e}

Il

such

lhat, r j-lceY c-i4. 'I'hereforc. -lr,ek

^and

taking into accoult

(12)) one obl,ains

irrf{ø'(a,

-y):yeyi ^{4ø'(n-sì- /¡) <}

^lll

^{È'lll ll-hll < ll-Lll}

for ail

ø' e

B". lahing the infimurn rvith

respect

to all

^fu

in Ií

such

that

n{lceY,ii

follor.r's

thatinffø'(n -y):y eYi (

E(a;), so 1,hat ,S(ø)

: :

suprinf{ø'(e;

- ^y);

^j/

^{€ y}}

^{: z' e}

B)} _ç

1l(ø).

Ðenote by epil? the cpigraph of the

fu^nction

D,

i.c,,

epi-0:

_{(ø,

_")e Z x ll:

^l?(a)

(

ø.1.

Sincc 14 is p-convex

(Ploposition

3.2),

its

epigraph

is a

^{p-convex subsel,}

of 7' X ll. By the

hypothesès

of the

theorem, there

is a point

zo

i-t Z

^at,

wlrich Z is contiluous.

\4/e

shall

show

that

(zo, fr@o)

11) ^is an inte'rior

. ^{3: Ilutlit¡'}

I'ehtfious

frrr best appruxirnutiorr'u'i{,h nunicirI

r.cstric-

lioils. rrct -r

bc' a r.ectot npcrcrì.

-\

^crti¿ò

in

_-\.

is

a

norrvoid stbsot r( oly

suclr

that ),'J{ c rl for all

À

)

^0.

rï ^yis

a subsrrt

of

a norrnccl spacc

x

¿urd

If is il

conc

in r,

tlenotci b.1' rlrr(

.,

^Y)

,-ll +

_10, oo l

thc

clistunòc fun,c- tiorr,

tlcfined

b¡'

(9) d¡;(,r,I):inf

{ll

r -Ìlll ^:ø

^e

T ilntl

ry

-- r

^e

tii:

:inf

_{ll

-hll:

^tt,

^{eI( ¿nrl n -l-tr e} Ii, r: eX;

rl¡dr,

^.Y)

is

calle,d

the

^l¡¿sl ctppl.o;.ri,nù(Lti,olù

o.f t; uítlt

con.ica,l, resl,r,ídi,on

I{

ltu

elent,ents irT,

I

(rvc aclopt ther convention

inf

Ø ₌₌ æ).

l!ìhe

plobleur oI the

cxistcuce

of

iur. r7,, i

rr I' u'ith 7o ^-

^{¿ e}

/l

such

tlrat cl¡('4 Y) ':

^ll :ü

- ^tloll

contains as

ptlrticular

cascil rnan.r'np¡rrorimir,-

tion

ploblerns

u'ith

lestr.ictions such ¿ls

thc

olrc-

the apploxirnation of a function ø

b.r' fur"Lctions

1or,ri(t) <

^n(¿))

for

¿ril ú

in a givon

iirter.val. -,\

these proltlorns

is

ttone

in

l1lì'1, Chap.

II,

rvhdib

s,l'stelnatically applitrcl

to obtain

t¡xact st'¡lutions for' 'r'arious appr,oxirnatiorr pro)llcms

u'ith

rcrstlictions

(espcciall¡'l'ith

lespeot

to an

intitg'ral utctr.ic)

l'ot'

some concl'eto classes

of flurrctions; the

consicle,rctl appr.oxirnatitr[' 1'uitctions

rru the

substrlace o1ì

algeblaic ol

trigonomrrbr,io iolyuonri:r,lsj

the

space

of

splirrc furrctions,

a1d the

^sc¡t

of fuñctions

fuar.ing"a rlegrec

of

srlootlrncrss highet,

than

bhe ¿p¡;¡6¡im¿¡t,e<1 function.

Irr

the duality

thcolenr plof¡en belou', r\-e supposc

ihal,

T

is

a, p-con-

t'¡:r

^.qc',t

and /f is

a convex conc. l['he

follon'ing propositiol

sholys

that

u'e gaìn nothing-

in

genelalit"r. supposing l,he cone

,I( onl¡.

p-couvcx,

l)tr,opo;ir.r:Io¡¡t 3.

L.

[trtt¡1.¡¡ _lp-cot¿pe

I

cone

is

^{coll,,L.e ;rt.}

,

l?¡'oo.f'.

\:T, I( ltc a

c<tnc'in

a

r.ecl,or spàcrì ,.u.

llhcn If is

convsx

if

anrl

onlr-

_{_il}

/f t I( 9 I{.

^}Ience, su1po,sing:

lløt I( is

p-conr.ex, \\.e have

to

_{lrrorr.e_}

K + K i-,K, lf

^irr

^aìld y ale in 7f,

thctn

¡r-il' ^and (I -

fr)-"1t

ale irr 1l too (rccall thLt 0 <? < ¹⁾

a,nd, thcr,efor.e, ,r

J

U

:.p2)-i,.,; i- + ⁽¹ - ?)(I - ^p)'1y el(.s

C)oncerning

the

r1i¡tanco

fulrclion

tlr;(.r'jf), \\rc pl.tlvc

llrlolosrtton

3.2,

I'et,Y

bc ct t¡,ornteil spcrce.

If I{

ís &co,n,Dctu cr¡n,e i.n

X cnttl Y'is

^cr' tt'on,ar¡icl'p-cotttert; su,bset o,f ,Y, ttietr tltc tlistant'e ftrtr.ctiott ^drr(.

, 'I) is

p-crn'ue,tt, i.e.,

{10)

rl¡;('1t,t +- (1

-

^,¡t)

^t:','J1)

^<i ptt{¡¡(tt:,

T) _l-

(1

--

¡t)

tl¡(et,'y)

.for ^ttl,l,

t:

ur¡cl

a' in,

_'Y.

... Proo.'Í'.

It is

suffioionl,

to ¡tloyo (10)

^{rvher-L dr,(a:,}

II) ilrtl

rlr,(a;,, T) alo

tinitc

nttmbtl's. Givc'n ^e

)_i),

thele cxist r7 tr,ntt ,u'it't

ll

such

that ,I

-

^nl,

'.t1'

- r'9.{:ll,t:-!tll {

^tl1¡(a;,

Y)f ranclll,r' - u'llttl¡¡(:u, Ii) i

^u. Tho'l;-con-

voxit¡' of Y

arril

the

convexitv of

1i

irrrpl.v ^,t)]J

+'(I - ^Ðlt,e Y anil _I)y

1_

+ ⁽¹ - ^p)y' -

^(p:t:

^{)- (1.-_/¡) ;'')} -

^pQl

- ^¡,t

^-F

¹ - ^fù(!t' - ^r/)

^e

I{"

^s,o

bhat ( pllr,

^dn(1t;t'

- ^llll -l

^{-l- (1}

⁽¹ - -- 'l)ll:t/ - ^p)

^r'.,

^Y) {

^,!t'll llpr,

< { tt

Ttct¡¡(t:,I)

--

^'p)

+(I - ^(n' -"Ø.ìì

1ì¡tJ,,1n',IZ)--¡'e.. As

-- Í -

^p)

^ú,)ll<

e

) 0 is at'bitlaly, incqualil,y (10) hoicls.

^g

7

102 gT. COEZAS and I. MUNa!ÐAN

Doiìtt of

epi ^É1.

To this

encl,

b¡"

l,he

continuity

"ottttttton, therc

cxists a

5->o

sncÌì

that _ilÌ(z) - ^{II(zòl <1-1: for ail}

^ø

^{in Z u'ith}

^{ll ã}

-

^Èoll

<

^ò.

Rernarh

th¿r,t

the

neighbtrurhoocl {ø

e7':llË -

^{zoll< S}}

x)lù(z)t*'-t

2

of flre point

^{(ao, I4(zò}

i ^{1) is included} in

epi,l4,

so_that

(øo,I)(zn\

t.L) ivill

^be

^{-in the} iite.ìiör. of

cpi -l?. Inrleocl,

if ^(ø,

^{ø) e}

Z x 1l ^is

^such

^that llz-zoll <ònnrtø )D(z) )-;-, ^tlrt'n

^-11(z)

-0(;o) 41 irn'Iicstltab

a

)>

Itr(z) -F ! )

^7)(øo)

I

^IJ(z)

-

^It)(øo)

:

It(z), z

ltcnccr

(ø,

o.) e cpi ^14'

îlre point (r,

D(u))

is

a l¡ouncta,r¡'point

of cpi

D becausc

(a, E(r))

^e

e opi -/1, a,nc1

if

V:{øeZ:lln-zll <rl x lE(r) - ^{e, It(r)} *'[' r ]0,

^e

]0;

is

a rrcighbourìroorl

ol'

(ø,

ü(r'))in i4x Il,

t'lrr:n (

\ r,

^I')(tt'\

- *)e ^{21 l/\.cpi}

^D'

Aulrlving

^{Thcolem 1.3,}

thele

exists a closedhype'rplanc

in Z

X

1l

suppor-

iäiä Ëinh

^{at, the}

point(ø, E(n)).This

^means that thereis (a*, ^{À) e} Z*

x ^Il : :"Ø x n)*, ^(z*',

^À)

I ^{(0,0) such}

^that,

(18)

^z*(n)

*'ì'' ^{Il(n) 2}

^ø*(ø)

* À'"

fol all øeZ

^{attd. a,ll ø e}

^{B rvilh}

^o-

)

^1l(z).

I1

^À

: 0, then

ø*(n) ^>- ø*(ø)

for all

z e

Z, irnplying^ that

^e*

:

0,

rvhich

contrad.icl,s

the

hypothesis ^1,ha1,

(z*,

À)

+ ^(0, 0). therofo.c, i. +

⁰

ancl

tahing ø: ⁿ in

^(18), one obtains

À'-Þ(a')

2 ^),'ø. e )'' _[Ð(ø) -

^æ)

) 0 for all ^o

^>-

^I](n),

rvhichimplies

^),

(0.

Dir.icting ir-rccFrality ⁽¹⁸⁾

by -).>>

⁰

andtlcrroting øf;:

: _-)'.-1 'a*,

one

obtlins

(1e) ^zt@\ - ^{It(*) >}

^z[(z)

-

^u

for all zeZ ^andall

^{ø e}

,ll ri'ith u à ^{E(ø). When} o:

^It)(z),

^inequality

(19) beoomes

(20)

^øl(n)

- ^{E(n) >}

^øt@)

-

^-ÉJ(a)

^for a[

^z

eZ'

To conclude the proof of ^Theorern

3.3,

rre !eed. ^t]re follos'i¡rg lemma,

wnich

appears

in tf¡lì ^p.

^38,

^{but our proof differs frorn}

^{l,he one}

^git'e'

therein.

Lnivrryr¡. 3.4.

If

ø' ^e

Z'

søftis/ies

llla'lll ^{4 L,}

^tlt'en

,(21)

^sup{a'(ø)

- ^{E(ø): zeZ\} -

sup'rø'(y):

y eY}'

Il

lllø'

lll >r

(includ,in'g tlta aasø

lllz'lll : æ),

tlt'en

,122)

^suP{ø'(a)

-Ð(z): zeZ}:oo'

P-CONVEX ^{SETS 103}

Proo.f

o.f'Lenima

3.4. Lei;

^B'

eZ' :villt ill¿'lli < 1, atrtl let

Y

el'"

fi'rorn

0 e

Il antl

.r7

f ⁰ - ^{ll e Y, it,} follou's that

0 < Ð(y):inf{lly -Qt ^+h)il : l;eíi,

ry -tr

keY) _{< liy} -7ll :0

rvhich yielt'ls lù(y)

-

⁰

^fol

^all

^{u eY.}

^flìhis trqualil;--v :rncl tht'ilrclusir'¡tt

ll c

Z plocluce

sup[ø'(ø)

- ^{IX(z): øeZj} )

^suple'(u)

- ^{I')(ti): y eY) --} :supiz'(y), ^ye1ll,.

¡61v, taliins into

^accounl,

tlr:Îiltition

(1G)

of

^S'

alrtl inctlnalit¡r

^(1?),

onc

obtains

ø'(ø)

-

^sup{e'(ø)

^{: y eY) <}

^B(e)

^{ç //(a) lo'- all}

^z

^eZ

giving tìre opposite ineclualit¡.

sup{a'(ø)

-

^IX(ø)

^:

^{z e Ø)'}

^<

srlpiz'(ll)

t _ll e

Y';, needetl

for thc plcof of

eclualitl' ^(21).

If _llia'lll >1,

i,henthele exists

n l;eZ n (-/í) l'itit _ll/ill :1

*.',"it tJnaí

ø'(lt:):1 + ^ø,

^where

^ø )0.

^{Sincc Ä;}

^eZ,

^iT'

^{follol's (by the}

^h.vpo-

theses

of 'lheolem iì.3) that, Il ¡

^(fu

I ^{fÐ + Ø,}

^{-se l'11¿¡}

^lhclo

^{arc l¡' e}

^I{

andyne Ysuch that,r7o-l,:J_lil

^ay

y,r-l¡:-l,feIí' [ìol' ttn¡' \>-7, the rclat,ion

^(À

-

¹⁾

^'(-lo)

^e

f{

irnptrie's 1'ha1, _J/o

* ^À(-/,') : _ïo -

^{l,: -r}

¡

^(À

- 1)(

^lt,)

^eI{

^1-

I{

^<-

lf

^(see

the proof of

i?r'oliosition

3.J).

'llhcle"

foLe,

Z(Àtu)

:inf

_illÀiu

- ^yll:

^y

^{€Y, !/} -

^À/¿e

^/r] < llÀtu

t/oll ^<

<

^Àll/r;ll

*

^llyoll

:

^ * lly.ll for all t 2

^f

.

Consequentl¡',

e'(f )'/') Ð()'tt:):

^À(1 -l-

a) -

^/l(),1;)

^{2 À(l -l-}

^o")

-

^ill/oll

-

^{À --'}

: À" -

^llyoil.

Since ^v"

) 0,

^À/c

e'/

^a;:rtl _-À/r; ^e

Z lor

t'.Il À

)

^1,

it follot's

^1,ha1,

sup[ø'(z-)

-]ì(z): ^øeØ) )sttpLø'(f],/i)

^//(À/i;)

:

^À

)1] :co.

Lernrna 3.+ is

^plovecl

.

$

Norv,

let

us continuo

thc

proof oli il'heolent 3.3.

\\¡e intencl to shol' that

the furrctional

af

cotrstructed above

is in

^/31.,

i.

e., l;lei"lll

< L

Supposing

the co trary,

ìll"å*lll

)J,

^ancl

^{using Lemrna l].,tr}

^¿llrrl

inequality (20), ive

^obtain

zt@)

-- ^n@) >

sup{air(ø)

- ^{E(z): r e'i!'} =:}

^co.

On the otht'r'

hancl,

by

hypothesesl

of l'heoletn 3.3,

^(o: -l- 7i) n

\: ₊

Ø,

so l,hnt

^Il(ar)

is a finite number'.'Iht¡ obtained.conlratliction

shorvsthal u'e musl, ha"¡.e lllød'lll

< ^[,

^{therefore relal,ion (21) of}

^the

^samc Lenììnà â]rd

Ínequality

^{(20) yielci :}

øl@)

--D(u) ^>

^sup[øf(ø)

-E(r): zeØ]: sutrizt(A)t y ev).

IJ

104 $'f, cctBzA$ ^al1d. l. MU,N1,.II:r.\N 10

Itronr this incqualit¡'

and inoqua,lity (17) onc tletivos

that

(23)

^,Y(er)

2

^cf

(r) -

sup {e$(v) ^:.r¡

e Y}

>

Il(t) ù ^8(x),

hencc,

(24)

II(:u)

: _ñ(c) -

^{ef (rr;)}

-

sup {ef (g) ^: u

eY}.

llo

^conclutle

^the

1ttool', ^tvcr lt¿lt'e

to

^shou'

that if

^rr e

Z\Y,

^then

llìeflll

-

^1.

^Iìy

^{(21), z{}

t

0 ^be¡caustt

n( I

irnplies

that

/4(.r)

Þ

^0.

\\¡o

hnow

itrat

_¡11a¡111

_< 1. If llleflll<r, ^then

lilÀeolll

:

1,

tthclo

¡.

: lll"Jll-')

^{1,, and}

reasonirrg

lilic in

the final

palt

of

theproof

oÎ illheolern 2.4 tvo

got a

^cotì-

tlaclicl,ion.

lhe ^prooli of

ilhcorcrrn

ll.il. is

ootnplct,e.

I

lìn¡r¡rln

3.5. Wlion

Z:I( :

-1, the d.istance

function cln(:,Y)

¿ìgÌeos

rvitlr the

usua,l clistanco

functiot

^d,(n,

Y) :

inT

{ll, - ^llll: I

^e

)i},

^{rr e}

't,

ant1, as it, is rvell ltnorvn,

this function

is continuous (in

fact it

is ovcrL

lrip-

sclritz, i.e., _ld(æ,

Y\ - tl(r', Y)l < llr -

^ar'll

^{tot' an\.} {t, ttì'itt -'li,

^see

l20l'

p.391 _). thelef.ore, lllteolern 3.3

crtends

Theoretns 2.3 ancl 2.4. .l.he func-

l,ional tt¡r(.,Y\ is not

^alu'a)-s

contimrous

as

is

shon-n b.\-

an

crxanrplcl

in

f131,

p.

10.

.l.he tollorvirrg erarnple

shot's that thclc exist

p-coLn

er

fulrctions ctefinecl orì p-co]t\¡ex scts

lvhich ale not

continuous on

the

$¡lìole domain

of

clefinition.

Iìx,trlr,n 3.6. Let X: llzecluippetl tvithtlte

Eucliclea,n

nornr

^a,ncl

lc¡t

Y - ^{I@,y) eIt2: løl -F lyl < 1ì u {(r,}

^{u) e}

Q":løl * lyl :11,

n'ìrere

Q

^tlcnotcs

thc

sel; ofi r'aticltr¿r,l nrunbet's.

il'ho fulrcticln.f : Y

^--+

ß,

definccl bJ'

./(r, ^!/) -

^ltul

+ lyl fol

_latl

* lyl < 1,

ancl

.f(n,y): ^{2 for}

\n,

^{y) e Q2}

\'ith irl *

^I

lll: ^{7, tt} _*-con\¡ex ^l¡ut it is continuous

only

oninl, I',:

l( :t',!ù

e Il,!:t¡l:l t ly¡-<

^{f l.}

lriho in thc

o¿rsc

of

best zr,pploxirnation

b¡'

clt¡lncnl,s

of a

_7r-convcx

sct

(Corollar'¡. 2.5), ^1'r'onl llhct¡r'crn

ll.ll

oììe

ciìn

delivc'

:lcharactclization

of

eleurc¡rrts

oÍ

besl, apptoxitna,tiott

I'ith conical rcstlicliorts.

Conol,lÀn,l' i\.1

. I'et,Ybe

u'not"nt,eil spuce,

let I(

be

(t

corl,rcfi cotte

in X,letZbeusubsltuceo.f X,ne Z:..Y

^tr,ite)'y

^e I¡(,r lI{), ult,et'eÍ'ís a

subset o.f

Z

sucl¡

that

Y n (z -[-

I() + Ø for

^a,Il

^{øeZ. In,}

ortler tlt'øt

q

^be

a

projection

o.f

æ

onto

y ¡(n + ^K),

^i,t

^is

suf.ficient a,ncl,

i,f Y is

p-conaeü, ,also ,necessary to e;nist

zi eZ'

wt'th, th,e proTterties :

a) lllz'' lll : I;

^{b) z; (.r}

-

-

^A)

:

_lln

-'yll;

^anc,l

c) zo'(U):

sup{z;

(U'): lt' eY). IJ Í is

^p-conaer,

then, l,he Junctional zo ca,n be clt'osen tr¡ l¡e continu'o'u,s on 7'.

Proof. Ißl zs be a functional in Z'

sa'l,isfying

ø), b) antl c). l'or

,every

y' e Y rvith y'

e

n +

^-K

the inequalit¡.

^(1.2) implies

lln -ll

^ll

:

zo

(r - ^u) : øi@ -

^u')

^{+ zi@') --}

^ø'

^(y)

^<

(

^zo

(ø - ^{y') < ltlø'olll llr} -

^{y' ll}

:

_llø

-

^u'll,

-which shows Lh.at

y is

a projection

of

ø

onto T

_{n(æ 1-}

I{),

Ciorryc¡sellr, suppose

tìrab l is

^p-convtìx

^antl let r/ lrc

¿l

plojection

of

^ar

onto ^T

n"('æ

+k¡- ^{B¡' lìheo.e;.} ll'i,, l;ltore exisl's r{ eZ*,

tvil"h lllzü il I

: _1,

such l'hat

\25)

^llr,

-

^{y ll}

:

eJ(æ)

-

^{sup{ø,f (u')}

^;

^{Y' e}

Í).

thc prooÏ rvill be

complcttr

it

^rvo

^shou' that

^{supføf (tl')}

^: !'€Yl:

: zi!).

^Othelrvise,

sup{4f;(17')

: y'e I-l >e;*(y) ^tr'r-rrl, since ⁿ -

^{Y e}

e

(-/l) ¡ Z, inequalit¡' ⁽¹²⁾

^vicltls

llr -yll : _{lllr#lll llø} - ^{ttll >}

^zfء)

^{-zi?) >þð(ø)} -

- sttl)',"t0') :

1¡' e

X), contlaclicting cqualit)'

^(25).

¡

/l. _Ilest upPrrlriluulion b¡' elt'nrt'lts

1¡[ s¡11r1'lts.

,\' subst:t ]i of

^â,

norrngcl space

i'is

callorl 1,-cctie'rrt,

if ^its

coruplc'tnt¡nt -\:

'\

_Y

is

a bouncletl p-sony6,x'set

s,ith

notrvoiil intt¡r'ior'. ilìhe

stutt¡'

o1' Lltrsl, allploxì.tlation_b-t'

älcmelts of

câvems (subsets

of

a

ltolrnctl

sp:rct' r,r'ith

_nonvoitl

^llounclecl

.convex

anrl ^tlc¡tti

^{a,trtl I.Singcr}

[61. The

pt'o

^{czÙI'erns was Posecl}

tt¡''ft. ta,r" still

lursolvstl

plo-

biern

of con s'

lllltc'

terrn

^ú(I{lce

'câvcrn"

^\yâs plollosetl ll-v

tì'

^Aspluncl [21.

Thc iiollowing tìreolorn cxtcntls

to

p-cÍr,r't'trts

the

^rnait-L

tlualit¡'l'esult in [6], lhcoretn

^2.1.

.llrrUOlnll [.L. Lat

,Y. be u, ttt¡t'ntatl ^SP6Icet

let Y

bt¿ tt p-cctttct'tt,

i'n X

cntil n

€f \Í.

^Then,

(26) inl{ll,r -yll:Ye Y}: ^inf [sup{r*(.t'):r'e-\:\Tì - -

^{t,i (,Î)}

:

^{¡r,"* e}

^5"i ^{.Ì I}

uherc,St :

_{ø{' ^e

_-\l* :

ll ,r* ll

: )t¡

'is tlLe u't¿it s'¡tltet'e 'in llte

tlttal

sltttce 'Y*

oÍ x.

Proo.f.

Put t/ : inf{ll

1,

-'llli ' y. {l ^{rx(r'.) i.}

^n'..e

e _'\,

\ ^yll -

^r*(,r,)

^:

^¿*

^{. Sol.}

^{i,q.¡ o''i' E}

¡{', sn¡r[,r'r(;')

^:

"' . .f \'Yi

^(c

is'l'irilc

bccaust¡ -X

\

^{Y is}

r

^tlcrl)'

H : _{{n' e} 'Y

^r'F(n')

: _!l

is-inclLrrlctl

in _-Í.

Intleetl,

iÎ n' 'eint(I \ Y). Since.I \ ^Y '- ^lel'E {;1r*('r")

_--(

< c) ^int(-Ií \ ^IJ-c int{ø"

^e

'!

.:

YiV")

^S

,}

=

^{n'-'-e

X=-"

**(á ^**('n') ì ^c.

Therefole, n'

f

^{11 showing}

^that H ^cY _'

B¡, Ascoli's formula

for the

distancc

from

a

point to

a hypelplane

in

:a normôd space (sec f201,

¡t'

24) rve have

¿:tl(n,'Y):¡t(t', ^y) :intLllr- yll:

^{y e}

7l <irfl'll't --yii:'yeHj:

: _lr*(ø) - ^cllllr*ll : sup{ø*(n'):n'

^e

X\ ^yi - ^**(n).

'lherefore,

,(27)

cZ

<infisup[n*(ø') i n'eX\Yi -n*(n)i ^nooeB*]:/'

?-CONVE)( SETS 105

11