ARKIV FOR MATEMATIK Band 4 nr 2
C o m m u n i c a t e d 12 N o v e m b e r 1958 b y LENNART CARLESON
The modulus o f c o n v e x i t y i n n o r m e d l i n e a r s p a c e s
B y Gi3TE NORDLANDER
W i t h 1 f i g u r e in t h e t e x t
The modulus of c o n v e x i t y 5 (e) of a n o r m e d linear space is defined as inf ( 1 - x + Y l ~
~(e)=
(Clarkson, D a y (1), L e m m a 5.1). F o r example, the modulus of convexity OE(~) of an a b s t r a c t Euclidean space is easily determined from the parallelogram identity a n d is found to be
die (e,) = 1 - r
d
E 2
4"
Theorem. The inequality
(e) < ~E (~)
holds /or every modulus o/ convexity r (e) in a spac~ with real scalars.
Proof. I t is sufficient to prove the theorem in t h e two-dimensional case. I n the usual w a y such a space m a y be t h o u g h t of as a plane in which the n o r m is determined b y a central-symmetric convex curve F with midpoint in O. The n o r m of a vector is determined b y the ratio of the length of the vector to the length of the parallel vector starting at O a n d ending at F. Thus F is the locus of the endpoints of the vectors from O of n o r m one.
If the two unit vectors x = OA a n d y = OB are r o t a t e d a r o u n d F, while their difference x - y = B A has c o n s t a n t l y the n o r m e, the endpoint M of the vector O M = 8 9 describes a curve F~ (Fig. 1).
l~ is a simple closed curve, as can be inferred from L e m m a 2.4 of D a y (1).
L e t Y be the area of the region inside 1 ~ a n d Y~ the area of the region inside F~.
L e m m a .
The lemma is a simple extension of a theorem of Holditch (de La Vall~e Poussin, p. 318), and is p r o v e d b y the same method.
15
G. :NORDLA.NDER, The modulus of convexity in normed linear spaces
C - . A
\ ,, .,,n / p
Fig. 1.
(The polar angle ~ is measured from an arbitrarily chosen direction.)
L e t (xl, Yl) a n d (x~, y~) b e t h e c o o r d i n a t e s of A a n d B in a n a r b i t r a r y rec- t a n g u l a r c o o r d i n a t e system. T h e n
Y = f yl d Xl ~ f y2 d x2
where t h e v a r i a b l e s xl, Yl, x2 a n d Y2 a r e t h o u g h t of as d e p e n d e n t of a p a r a - m e t e r t, v a r y i n g f r o m ~ t o /~ a s t h e curve is d e s c r i b e d once in t h e n e g a t i v e direction. The c o o r d i n a t e s of M are t h e n ' xl§ Yl~-Y~ a n d
[
\
2 ' 2 ]Y~ = 88 f (y, § y~) d (x, § x~).
H e n c e Y - Y~ = ~ f (YI - Y~) d (X 1 - - X2).
The i n t e g r a l in t h e r i g h t m e m b e r expresses t h e a r e a of t h e region inside t h e curve d e s c r i b e d b y t h e e n d p o i n t of t h e v e c t o r AB, if t h i s v e c t o r is l a i d off from a fixed p o i n t . This curve is similar to F in t h e r a t i o e. H e n c e
Y - Y~= 88 Y, which p r o v e s t h e l e m m a .
The n o r m of t h e p a r t MC of t h e u n i t v e c t o r OC t h r o u g h M is d e n o t e d A (e, ~).
W e h a v e
(~ (e) = inf A (s, ~).
r
If t h e areas Y a n d :Y, are e x p r e s s e d in p o l a r c o o r d i n a t e s t h e l e m m a t a k e s t h e f o r m 16
ARKIV FOR MATEMATIK. Bd 4 nr 2
2~ 2~
o 0
where r(~) is the l e n g t h of the vector OC. Hence
which implies
because r 2 (~) > 0.
2~
0
V ~2
b(e)=infA(e,~)~<l- 1-~=~E(e),
Theorem 4.1 of D a y (2) shows t h a t if there is e q u a l i t y i n our t h e o r e m for all e, 0 < ~ ~< 2, the space is a b s t r a c t E u c l i d e a n . I have n o t been able to decide whether the same conclusion holds u n d e r the sole a s s u m p t i o n of e q u a l i t y for a certain ~0, 0 < e o < 2.
R E F E R E N C E S
CLARKSO~-, J . A . : U n i f o r m l y c o n v e x spaces. T r a n s . A m e r . M a t h . Soc. 40, 396-414 (1936).
DAY, M. M.: (1) U n i f o r m c o n v e x i t y in f a c t o r a n d c o n j u g a t e spaces. A n n a l s of M a t h . (2), 45, 375-385 (1944).
- - - - (2) S o m e c h a r a c t e r i z a t i o n s of i n n e r - p r o d u c t spaces. T r a n s . A m e r . M a t h . Soc. 62, 320-337 (1947).
DE LA VALL~E POUSSI~r CH.-J.: Cours d ' A n a l y s e I n f i n i t 6 s i m a l e , I.
Tryck~ den 2 upril 1959
Uppsala 1959. Almqvist & Wiksells Boktryckeri AB
i t 17
