数理解析研究所講究録 1246 巻 2002 年 93-97

93

$q$

On some generalizations of-uniform convexity inequalities

九州工業大学工学部
岡山県立大学情報工学部
広島女学院大生活科学部

加藤幹雄 (Mikio Kato)
高橋泰嗣 (Yasuji Takahashi)

橋本一夫 (Kazuo Hashimoto)

Abstract. This is an announcement of some recent results of theauthors concerning the -uniform convexity and -uniform smoothness
inequalities.
$p$

$q$

We shall consider some generalizations of -uniform smoothness and q-uniform
convexity inequalities. In particular we shall characterize these two geometricnotions by tyPe- and cotype-like inequalities which are stronger than those of tyPeand cotype, respectively.
$p$

1. puniformly smooth and -uniformly convex spaces$q$

Let $X$ be aBanach space with $\dim X\geq 2$ . The modulus of convexity of $X$ is$\delta_{X}(\epsilon)=\inf\{1-\frac{||x+y||}{2}$:$||x||=||y||=1$ , $||x-y||=\epsilon\}$ ,$0\leq\in$ $\leq 2$.

for all $>0$ , and -uniformly convex
is called uniformly convex if$(2\leq q<\infty)$ if there exists aconstant $C>0$ such thatfor all $>0$ .
The modulus of smoothness of $X$ is$X$$\delta_{X}(\epsilon)>0$$q$$\epsilon$$\delta_{X}(\epsilon)\geq C\epsilon^{q}$$\rho_{X}(\tau)=\sup\{\frac{||x+\tau y||+||x-\tau y||}{2}-1$:$||x||=||y||=1\}$ ,$\epsilon$$\tau>0$.

, and -uniformly smoothasis called uniformly smooth if$(1 <p\leq 2)$ if there exists aconstant $K>0$ such thatfor all $\tau>0$ .
These moduli have the best values with aHilbert space $H$ (cf. [8, p. 68]): For any
Banach space $X$$X$$\rho_{X}(\tau)/\tauarrow 0$$\tauarrow 0$$p$$\rho_{X}(\tau)\leq K\tau^{p}$,$\delta_{X}(\in)$$\leq$$\delta_{H}(\epsilon)=1-\sqrt{1-\epsilon^{2}}/4$$\rho_{X}(\tau)$$\geq$$\rho_{H}(\tau)=\sqrt{1+\tau^{2}}-1$.

In view of these facts no Banach space is -uniformly convex for $q<2$ and punifomly smooth for $p>2$ . In fact, if $q<2$ ,$q$$\sin \mathrm{c}$$\frac{\delta_{X}(\epsilon)}{\epsilon^{q}}\leq\frac{1-\sqrt{1-\epsilon^{2}/4}}{\epsilon^{q}}=\frac{\epsilon^{2-q}}{4(1+\sqrt{1-\epsilon^{2}/4})}$,

94
we have. When $p>2$ ,$\lim_{\epsilonarrow+0}\delta_{X}(\epsilon)/\epsilon^{q}=0$$\frac{\rho_{X}(\tau)}{\tau^{p}}\geq\frac{\sqrt{1+\tau^{2}}-1}{\tau^{p}}=\frac{1}{\tau^{p-2}(\sqrt{1+\tau^{2}}+1)}arrow\infty$as$\tauarrow 0$.

for all $\tau>0$ . It is
Also every Banach space is 1-uniformly smooth asclear that -uniformly smooth spaces are -uniformly smooth if $1<r\leq p\leq 2$ ,and -uniformly convex spaces are -uniformly convex if $2\leq q\leq r<\infty$ .
$\rho_{X}(\tau)\leq\tau$

$r$

$p$

$r$

$q$

puniformly smooth and -uniformly convex spaces are characterized by thefollowing -unifom smoothness and -unifom conveity inequalities:
$q$

$p$

$q$

Lemma 1([1], [2]). (i) Let $1<p\leq 2$ . Then
only if there exists $K>0$ such that(1)$\mathrm{X}$$\frac{||x+y||^{p}+||x-y||^{p}}{2}\leq||x||^{p}+||Ky||^{p}$is -unifomly smooth if and$p$

for au , $y\in X$ .
$x$(ii) Let $2\leq q<\infty$ . Then $X$ is (/-uniformly convex if$C>0$ such that(2)$\frac{||x+y||^{q}+||x-y||^{q}}{2}\geq||x||^{q}+||Cy||^{q}$$\mathrm{m}\mathrm{d}$only if there exists

for all , $y\in X$ .
$x$

Remark 1. (i) The validity of the inequality (1) implies $K\geq 1$ . Thus (1)with the best constant $K=1$ is the following Clarkson inequality(3)$(1<p\leq 2)$$( \frac{||x+y||^{p}+||x-y||^{p}}{2})^{1/p}\leq(||x||^{p}+||y||^{p})^{1/p}$(ii) In (2) we have necessarily $0<C\leq 1$ (indeed put $x=0$ ), and the inequality(2) with the best constant $C=1$ is the following Clarkson inequality(4)$( \frac{||x+y||^{q}+||x-y||^{q}}{2})^{1/q}\geq(||x||^{q}+||y||^{q})^{1/q}$$(2\leq q<\infty)$

2. Generalizations of -uniform smoothness and -uniform convexity
inequalities
$p$

$q$

We shall present some generalizations of puniform smoothness and q-uniform
convexity inequalities which hold to characterize these smoothness and convexity.
More prescisely, in the first sense we shall give tw0-element inequalities sharper
than (1) and (2) respectively, and in the secondary sense we shall characterize-uniform smoothness and -uniform convexity by type-, cotype like inequalities
which are stronger than type, cotype inequalities respectively

$p$

$q$

95The notions of tyPe and cotype were introduced by(cf.
[9]) in the context of the law of large numbers for random variables with values
in aBanach space. ABanach space X is called of type p, 1p 2, if there is$>0$
M(necessarily M 1) such that$\mathrm{H}\mathrm{o}\mathrm{f}\mathrm{f}^{\ovalbox{\tt\small REJECT}}\mathrm{m}\mathrm{a}\mathrm{n}- \mathrm{J}\langle 5\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}[3]$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$(5)$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{n}\theta_{j}x_{j}||^{p})^{1/p}\leq M(\sum_{j=1}^{n}||x_{j}||^{p})^{1/p}$

for all finite systems , \cdots , $x_{n}\in X$ . X is called of cotype q, $2\leq q<\infty$ , if there) such thatis M $>0$ (necessarily M$x_{1}$$\geq 1$(6)$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{q}\theta_{j}x_{j}||^{q})^{1/q}\geq\frac{1}{M}(\sum_{j=1}^{n}||x_{j}||^{q})^{1/q}$for all finite systems$x_{1}$, \cdots , $x_{n}\in X$ .

These probabilistic properties are characterized by Clarkson’s inequalities which
are of geometric nature. Namely, in 1997 the first and second authors [6] showedthat $X$ is of type with $M=1$ if and only if Clarkson’s inequality (3) holds
in $X$ and the corresponding fact for cotype and Clarkson’s inequality (4) (theirpresentations are more general). On the other hand it is well known that$p$(i) -uniformly smooth spaces are of type ,(ii) -uniformly convex spaces are of cotype
$p$

$p$

$q$

$q$

.and there is no converse of these assertions. Indeed there exisits a non-reflexive
space $X$ having type 2(James [4]). Then $X$ is of type for any $1<p\leq 2$ , whereas$X$ is not puniformly smooth because uniformly smooth spaces must be reflexive.
Also its dual space
is of cotype for any $2\leq q<\infty$ , but not q-uniformly
convex as
is not reflexive.
$p$$X^{*}$$q$$X^{*}$

Theorem 1( -uniform smoothness). Let $1<p\leq 2$ and $1\leq s<\infty$ . The
following are equivalent.
(i) $X$ is -uniformly smooth(ii) There exists $K\geq 1$ such that$\mathrm{p}$

$p$

(7)

$( \frac{||x+y||^{s}+||x-y||^{s}}{2})^{1/s}\leq(||x||^{p}+||Ky||^{p})^{1/p}$$\forall x$, y $\in X$ .

If p $\leq s<\infty$ , in addition:(iii) There exists Ksuch that$\geq 1$(8)$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{n}\theta_{j^{X}j||^{s})^{1/s}\leq}(||x_{1}||^{p}+\sum_{j=2}^{n}||Kx_{j}||^{p})^{1/p}$

for all finite systems$x_{1}$, \cdots , $x_{n}\in X$ .

96Remark 2. (i) The inequality (7) is sharper than (1) of Lemma 1if p
Indeed in this case by Lemma 2s.

$\ovalbox{\tt\small REJECT}$.

$( \frac{||x+y||^{p}+||x-y||^{p}}{2})^{1/p}\leq(\frac{||x+y||^{s}+||x-y||^{s}}{2})^{1/s}\leq(||x||^{p}+||Ky||^{p})^{1/p}$(ii) For the case $K=1$ the equivalence of the inequalities (7) and (8) is provedin KatO-Takahashi [6].
(iii) The inequality (8) is stronger than the tyPe inequality (5). Indeed, thespace $X$ of James stated above is of type , whereas (8) fails to hold in $X$ . So werefer to (8) as strong tyPe inequality.
$p$

$p$

$p$

Theorem 2( -uniform convexity). Let $2\leq q<\infty$ and $1<t\leq\infty$ . The
following are equivalent.
(i) $X$ is -uniformly convex.
(ii) There exists $0<C\leq 1$ such that
$q$

$q$

(9)

$( \frac{||x+y||^{t}+||x-y||^{t}}{2})^{1/t}\geq(||x||^{q}+||Cy||^{q})^{1/q}$$\forall x$, y $\in X$ .

If $1<t\leq q$ , in addition:(iii) There exists $0<C\leq 1$ such that(10)$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{n}\theta_{\mathrm{j}}x_{j}||^{t})^{1/t}\geq(||x_{1}||^{q}+\sum_{j=2}^{n}||Cx_{j}||^{q})^{1/q}$

for all finite systems$x_{1}$, \cdots , $x_{n}\in X$ .

Remark 3. (i) The inequality (9) is sharper than (2) of Lemma 1if q
Indeed we have$\geq t$$( \frac{||x+y||^{q}+||x-y||^{q}}{2})^{1/q}\geq(\frac{||x+y||^{t}+||x-y||^{t}}{2})^{1/t}\geq(||x||^{q}+||Cy||^{q})^{1/q}$.

.

(ii) For the case $C=1$ the equivalence of the inequalities (9) and (10) is provedin KatO-Takahashi [6].
(iii) The inequality (10) is stronger than the cotype inequality (6). Indeedof the space $X$ of James is of cotype , but (10) fails to hold in
the dual spaceX.
is also acounter example, since it is of cotype 2and non-reflexive. So werefer to (10) as strong cotype inequality.
$q$$X^{*}$$q$$L_{1}$$q$

is of cotype , where $1/p+1/q=$
It is well known that if $X$ is of type , thenhas cotype 2,, and the converse is not true ([2, pp. 309-310]). Indeed,
whereas has no non-trivial type. Our next theorem asserts that for our strong
type and cotype inequalities (8) and (10) the converse is also true if $p\leq s<\infty$ .
$p$$X^{*}$$q$$l_{1}=(c_{0})^{*}$$1$$c\mathit{0}$

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$1$

Theorem 3(duality). Let $1\leq p\leq 2,1<s<\infty$ and $1/p+1/q=1/s+1/t=$. Let $1\leq K<.\infty$ . Then if(8)$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{n}\theta_{j^{X}j||^{s})^{1/s}\leq}(||x_{1}||^{p}+\sum_{j=2}^{n}||Kx_{j}||^{p})^{1/p}$

holds in $X$ ,$(10^{*})$

holds in$( \frac{1}{2^{n}}\sum_{\theta_{j}=\pm 1}||\sum_{j=1}^{n}\theta_{j}x_{j}^{*}||^{t})^{1/t}\geq^{\mathrm{t}}(||x_{1}^{*}||^{q}+\sum_{j=2}^{n}||K^{\neg 1}x_{j}^{*}||^{q})^{1/q}$$X^{*}$. If $p\leq s<\infty$ the converse is true.

References[1] K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness
inequalities for trace norms, Invent. Math. 115 (1994), 463-482.
[2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry 2nd Ed., 1985.
[3] J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186.
[4] R. C. James, Nonreflexive spaces of type 2, Israel J. Math. 30 (1978), 1-13.
[5] M. Kato, L. E. Persson and Y. Takahashi, Clarkson tyPe inequalities and theirrelations to the concepts of type and cotype, Collect. Math. 51 (2000), 327-346.
[6] M. Kato and Y. Takahashi, Type, cotype constants and Clarkson’s inequalities forBanach spaces, Math. Nachr. 186 (1997), 187-196.
[7] M. Kato and Y. Takahashi, Some recent results in the Banach space geometry andrelated norm inequalities, Publ. Dept. Anal. Mat. Univ. Complutense de Madrid46 (2000), 105-122.
[8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, 1979.
[9] B. Maurey and G. Pisier, Series de variables aleatoires vectorielles independanteset proprietes geometriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
[10] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20(1975), 26-350.
[11] Y. Takahashi and M. Kato, Clarkson and Random Clarkson inequalities for $L_{r}(X)$ ,Math. Nachr. 188 (1997), 341-348.
Department of Mathematics
Kyushu Institute of Technology
Kitakyushu 804-8550, Japan

Department of System Engineering
Okayama Prefectural University
Soja 719-1197, Japan
Department of Enviromental Health Science
Hiroshima Jogakuin University
Hiroshima 732-0063, JaPa