On moduli of convexity in Banach spaces
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Let X be a normed linear space, x ∈ X an element of norm one, and ε > 0 and δ(x,ε) the local modulus of convexity of X. We denote by ρ(x,ε) the greatest ρ ≥ 0 such that for each closed linear subspace M of X the quotient mapping Q : X → X/M maps the open ε-neighbourhood of x in U onto a set containing the open ρ-neighbourhood of Q(x) in Q(U). It is known that ρ(x,ε) ≥ (2/3)δ(x,ε). We prove that there is no universal constant C such that ρ(x,ε) ≤ Cδ(x,ε), however, such a constant C exists within the class of Hilbert spaces X. If X is a Hilbert space with dimX ≥ 2, then ρ(x,ε) = ε2 /2. 1. Introduction Let X be a real normed linear space of dimension dim X ≥ 1 and let U be the closed unit ball of X. Let ε > 0. The modulus of local convexity δ(x,ε), where x ∈ U, is defined by
x + y : y ∈ U, x − y ≥ ε 2
δ(x,ε) = inf 1 −
(1.1)
and the modulus of convexity is
δ(ε) = inf δ(x,ε) : x ∈ U .
(1.2)
If dimX ≥ 2, one can use an equivalent definition (see, e.g., [1]),
x + y : x, y ∈ X, x = y = 1, x − y = ε 2
δ(ε) = inf 1 −
(1.3)
and if x = 1,
x + y : y ∈ X, y = 1, x − y = ε . 2
δ(x,ε) = inf 1 −
(1.4)
The space X is said to be uniformly convex (locally uniformly convex) if for each ε > 0, δ(ε) > 0 (δ(x,ε) > 0 for x ∈ U, resp.). Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 423–433 DOI: 10.1155/JIA.2005.423
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On moduli of convexity in Banach spaces
The moduli δ(ε) of the spaces L p (µ) have been found in [2]; they behave for ε → 0 as (p − 1)ε2 /8 + o(ε2 ) when 1 < p ≤ 2, and as p−1 (ε/2) p + o(ε p ) when 2 < p < ∞. In case of a Hilbert space X with dimX ≥ 2, δ(ε) = 1 − (1 − ε2 /4)1/2 for ε ∈ (0,2]. We denote by ᐀ the family of the canonical quotient maps Q : X → X/M, where M ranges over all closed linear subspaces of X. For any ε > 0 and x ∈ U, let ρ(x,ε) = sup{r : r ≥ 0 and for each Q ∈ ᐀, Q maps the open ε-neighbourhood of x in U onto a set containing the open r-neighbourhood of Q(x) in Q(U)}, and let ρ(ε) be defined by
ρ(ε) = inf ρ(x,ε) : x ∈ U .
(1.5)
We note that if T is an open linear mapping from X onto a normed linear space Y such that T −1 (0) is closed and T(U) contains a c-neighbourhood of 0 in Y , then for each x ∈ U and ε > 0, T maps the ε-neighbourhood of x in U onto a set containing the cρ(x,ε)neighbourhood of T(x) in T(U). Thus the “ρ-moduli” help to estimate relative openness of T on U in a quantitative way. Relative openness of affine maps on convex sets has been treated in literature in various contexts, a list of references is presented in [3]. For each ε > 0, the following holds [3]: 2 ρ(x,ε) ≥ δ(x,ε) for each x of norm one, 3 2 ρ(ε) ≥ δ(ε), 3 4 ρ(x,ε) ≤ δ(x,λε) for each x ∈ U and λ ∈ (1,3], λ−1 4 ρ(ε) ≤ δ(λε) for each λ ∈ (1,3]. λ−1
(1.6) (1.7) (1.8) (1.9)
These relations suggest the following questions. Question 1.1. Is there a constant c1 such that ρ(x,ε) ≤ c1 δ(x,ε)
(1.10)
for all X, x ∈ X of norm one, and ε ∈ (0,2]? Question 1.2. Is there a
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