then the operator L is regular. Proof..
Let
x, / ~ L
~,
Lx=L Applying Fourier transforms to both sides of Eq.
(15) ...
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then the operator L is regular. Proof..
Let
x, / ~ L
~,
Lx=L Applying Fourier transforms to both sides of Eq.
(15) (15), we obtain
~(~) = ~'i'~ (~o) -- Ao)-* Aj~ (~ -- ~3) + ( ~ -- Ao) -~ ] (~) (fl is the Fourier transform of the function u). obtain the estimate
[I X I{L'
0 if t > 0. Then M is called an Orlich function. If M satisfies the condition ]im M (2z)/M ( x ) < q-co, then M is said to satisfy the h=-condition at 0. We note that if an x-+0
Orlich function M satisfies the A=-condition
sup M ( 2 x ) / M ( z ) < + o o
at zero, then
for any
0 < x-~K
number K < +~. Definition. Let M be an Orlich function. Consider ZM, the space of all real numerical sequences x = (xi)i 0 : ~ ' ~ 1 M (I x, l/t) < i}. Then lM w i t h t h i s n o r m i s c a l l e d c a l l t h i s an O r l i c h s p a c e ) .
a space
of Orlich sequences
( h e n c e f o r w a r d we s h a l l
simply
O r l i c h s p a c e s a r e Banach s p a c e s . M o r e o v e r , i f an O r l i c h f u n c t i o n M s a t i s f i e s t h e h=c o n d i t i o n , t h e n t h e s e q u e n c e o f v e c t o r s (e~)~.L/n,
M(x)-~ ~.
sup
0 < xi
0 there exists Proof. number m and real numbers x:, . ., x m in [0, !] (not all zero) such that
a natural
(I) Consider two cases. I)
Let
~=~ :~l (x0 > We may assume that X =
(xi)i~<mis
(2)
I.
a nonincreasing
sequence,
Consider the natural number n and the finite sequence fying the conditions
k0 = 0 ,
of such an n and
of natural numbers
(ki)i ~=~ M (x~.0 > lUll = 2. This.contradicts inequality (3). We now r e c o n s t r u c t t h e s e t s Yo, - - , Yn-~. Take Yo. I f qo < l , c h o o s e (2 -- qo) e l e m e n t s i n Z. D e n o t e them by yi . . . . . y~_~o. Set Yo = Yo [J {Yl. . . . .
y~_qo}. We
note that these always exist,
since
Z contains
~_~(l--q~)
elements,
and
q~
1
for
any i ~ n--i. If qo = l, we do not change the set Yo but set Y" = Yo and go straight to Y~. y' Consider the set Zo=Z~{yi ..... y~-qo}if ~ 0 ~ l , and Zo = Z if qo = ~- We construct z in exactly the same way as we constructed Yo, where Z is replaced by Zo, etc. Thus we obtain sets Yo ..... Yi~-i such that n--I
! and each Yi contains exactly I elements. It is easily seen that for any f ~ n - - I
We rewrite the elements we have the inequality
-~'~=iM (~) < ~. In fact,
308
let Y~ = {zll..... Xlqi, zl..... s(~_qi~), where
zj ~ Z.
! of Yi as uij
(] ~ l).
(4) Then
M(z) ~ 2/l
( h e r e we a r e u s i n g t h e i n e q u a l i t y (1) l o o k s l i k e t h i s :
for
,-o - , = . From (4) and (5) we see immediately
0, t o o b t a i n a c o n t r a d i c t i o n . t h e s e t o f a l l t h o s e i ~ n - - i f o r which
Denote by A
~zi=1 i (uij) > t/i0.
(7)
Then {A{> n/8 ( h e r e {A l d e n o t e s t h e c a r d i n a l i t y o f t h e s e t A). n/8. We e s t i m a t e t h e f o l l o w i n g sums from above:
Suppose t h e o p p o s i t e :
{A{
0 such t h a t
Here P depends only on the function M.
We now take i0 ~
21=1U~:] = min {~i=l ui~: Since M is monotonic,
[n18] -- i such that
i -~< [ n / 8 ] - i}.
it follows that
M(V~':o /'.
2j:lu~i)/([nlS]l)) >
/-~[n/8]--I
.
M(I//(j=lU~])/I) 2'
~
(12)
S i n c e t/iO.~ti=lM(Uw) 0, we can choose a natural
Set
number N s u c h t h a t N t ~ l , (Nq-l) w h e r e u i j = x i f o r any ] ~ N q - i .
2
Pl/P < ~e.
and bearing in mind (6), we can write ciently small, case I) does not occur.
II)
l
t > I.
Thus,
I < (N q- I) t < 2. Now take a family (u~l)i<m,~ 0, contradicts ~
the statement
o f t h e lemma and t h e
N+I
Thus Lemma i is proved. LEMMA 2. l)
Let M be an Orlich function.
The following statements
are equivalent:
lie (c O, lM) = L (c o, lM),
2) t h e r e e x i s t s a r e a l number C > 0 s u c h t h a t numbers %ij (i ~ m, ] ~ n), s a t i s f y i n g t h e f o l l o w i n g 0<X~/~t
f o r any n a t u r a l relations
numbers m and n and r e a l
(i~m,i~n),
(16)
i . ~ j = I M ( X ~ j ) < 2,
(17)
we have the inequality
Proof. We first prove that condition 2) in Proposition i is equivalent to the following condition 2') : 2') There exists a number K > 0 such that for any natural numbers m and n and for any normed vectors x~ ~ X n (i = I ..... m) , we have the inequality tr-~
to the inequality
(20)
~..
If we prove that (21) then using the convexity of M, we immediately obtain then for any i ~ m
(20).
Since [] x~ ][ = i (i----I..... m),
~ i i (Ie~ (x01) = I. Then to prove
(21), it is sufficient
Conversely,
to use statement 2) of Lemma 2.
let [Is (co, IM) = L (co, IM). Then to prove 2) of Lemma 2, it remains to use
Proposition I, taking
x i = ~'Y=1%r
= i .... , m).
Lemma 2 is proved.
Proof of the Theorem. i) Let II~ (Co, IM) = L (co, IM). We prove that the function M x (x) = M (V~ is quasiconcave, suppose not. Then from Lemma I, for any e > 0 there exists a natural
number
m and
0~
xt ~ I (i ----1 ..... ra),satisfying the inequality
i~.ilJ}l(x0~2,
for
which M ( V ( ~ . : i x ~ ) / m ) < elm. Consider now the matrix of real numbers (%ij)~<m.j<m, setting Xil = xj, and whose remaining rows are c y c l i c permutations of the first row. Then
i -.~ ~.~IM(~0)..< 2
(i = 1 , . . . . m)
and
which contradicts
Lemma 2, since ~ > 0 was arbitrary.
2) Now let 7~11(x) = M ( ~ f - ~ we use Le=~na 2.
be quasiconcave.
To prove the equation ll~(c0, I M ) =
L (co, IM)
Consider m, n and the set of numbers (%U)~<m. j 0 be the constant in the definition of quasiconcavity of M~. Then
which completes the proof of the theorem. COROLLARY i. Let M be a differentiable Orlich function with the A2-condition, N(x) = M'(x)/x be a decreasing function on (0, i ] . Then ll~(co, i M ) = L ( c o , IM).
and let
Proof. C o n s i d e r M~ (x) = M ( V x ) (x E (0,tl). Then t h e f u n c t i o n M~(x) = M ' (Fx)/(2Vx) is monotonically decreasing. Hence it follows that the function MI is concave on (0, i], and therefore also quasiconcave on this interval, a n d it only remains now to apply the theorem. To prove the following corollaries we need some definitions and results. Definition.
Let
p > 2
and X be a Banach space.
p r o p e r t y if for any unconditionally
convergent series
X is said to satisfy the Orlich p~,:=ix~ in X, the numerical series 311
~=i 11x~ li p
also converges.
Definition. Let p ~ 2. The Banach space X is called a space of cotype p if there exists a number C > 0 such that for any natural number n and any x j ~ X ( ] = i ..... n) we have the inequality
(i~l]E';=l rj(t)xj
~t) '/'>C( Y?j=l
Here r] (t) = sign sin (2i-1 2=t) i s the Rademakher f u n c t i o n . thus :
\lip
IIxj IIp)
.
Inequality
(22) (22) can be rewritten
I t i s w e l l k n o ~ (see, e . g . , [ 2 ] ) t h a t i f the e q u a t i o n E=(c=, X) = L ( c o , X) h o l d s f o r the Banach space X, then X s a t i s f i e s the O r l i c h 2 - p r o p e r t y . The f o l l o w i n g s t a t e m e n t a l s o h o l d s (see [3] ). Proposition. Let X be a Banach space with an unconditional basis. K2(co, X) = L(co, X) holds if and only if X is a space of cotype 2.
Then the equation
Using these fffcts, we obtain the following corollaries of the theorem. COROLLARY 2. Let M be an Orlich function satisfying the A2-condition, ~/i (x)= 7~f (~). If M~ is quasiconcave, then the space 1 M satisfies the Orlich 2-property. COROLLARY 3. Let M be an Orlich function with the A=-condition. of cotype 2 if and only if the function M~ is quasiconcave.
Then 1M is a space
The author thanks B. M. Makarov for his help with this article. LITERATURE CITED i. 2. 3.
A. Pietsch, "Absolute p-summierende Abbildungen in normierten Raumen," Stud. Math., 28, 333-353 (1967). E. Dubinsky, A. Pelczyncki, and H. P. Rosenthal, "On Banach spaces X for which I]2(L~, X) = B (L~, XJ," Stud. Math., 44, '617-648 (1972). B. Maurey, "Theorems de factorisation pour les operateurs lineaires a valeurs dans les espaces ~ , " Asterisque, ii, 1-194 (1974).
PROPERTIES OF SYSTEMS OF n-DII~NSIONAL CONVEX SETS IN FINITEDIMENSIONAL LINEAR SPACES A. G. Netrebin
In this article we consider combinatorial properties of systems of convex sets in a linear space. In Sec. 1 we prove Theorems 1-3, which are analogs of theorems of De Santis [i], GrHnbaum and Katchalski [2, 3], and we introduce an analog of theorems of Hadwiger and Debrunner [4]. The theorems of these authors establish several properties of systems of convex sets in the space R n. Theorems I, 2, and 4 state that these properties (with certain changes) hold for systems of n-dimensional convex sets in a finite-dimensional space. We also prove (Theorem 5) a generalization of Boltyanskii's and Soltan's theorem [5]. In Sec. ~ 2 we use several statements from Sec. 1 to study the nerves of systems of n-dimensional convex sets in the space R d. Standard terms and notation will be used. i. To prove Theorems 1-4 we need the following two theorems. The first of these (A) was proved by De Santis [i], and the second (Theorem B) was proved partly by GrHnbaum [2] and finally by Katchalski [3]. Institute of Mathematics and Mechanics, Ural Science Center, Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 25, No. 4, pp. 603-618, April, 1979. Original article submitted November 22, 1977.
312
0001-4346/79/2534- 0312507.50 9 1979 Plenum Publishing Corporation