Uniformly convex spaces

By J. Clarkson. 1936

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Variance-based Regularization with Convex Objectives (2016, neutral): “suniformlyconvexifforeachǫ > 0,thereisaδ > 0suchthat k x −y k ≥2ǫ implies 1 2 ( + ) 1 δ whenkxk =kyk =1. Projections ontouniformly convexnormballs areunique;p-norms,p ∈ (1,∞),areuniformly convex[12]. 13 AssumptionB. There exists a measurable function L : X → R+ such that for each x ∈ X, ℓ(·,x)”
A semi-invertible operator Oseledets theorem (2011, background): “ kfkHt p := kF−1(a tF(f))kLp is a Banach space known as (local) fractional Sobolev space. Thus, Jt is a surjective isometry from Lp(R) to Ht p. Since Lp(R) is separable, reﬂexive and uniformly convex [5] (see Deﬁnition B.21), and Ht p is the isometric image of Lp(R), it is also separable, reﬂexive a”
On a new geometric constant related to the von Neumann–Jordan constant (2006, unknown): “Recall Clarkson inequality [3,9], that when p 2, ‖x + y‖p + ‖x − y‖p (‖x‖ + ‖y‖)p + ∣∣‖x‖ − ‖y‖∣∣p, x, y ∈ p.”
Projection-Free Optimization on Uniformly Convex Sets (2020, background): “(Clarkson, 1936; Boas Jr, 1940) deﬁne a uniformly convex normed space ( X , || · || ) as a normed space such that, for each (cid:15) > 0 , there is a δ > 0 such that if x and y are unit vectors in X with || x − y || ≥ (cid:15) , then ( x + y ) / 2 has norm lesser or equal to 1 − δ .”
Proof Mining: A Systematic Way of Analysing Proofs in Mathematics (2002, background): “The notion of uniform convexity was introduced in 1936 by Clarkson [20] (see also [61]) and plays a crucial role in many parts of functional analysis.”
Axiomatic theory of Sobolev spaces (2001, unknown): “[7], [25] or [26]).”
On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces (2012, background): “On the other hand, Day [37] proved that a normed linear space X is an inner product space if and only if the Clarkson modulus of the convexity of X (see [35]), namely”
Rigidity of commensurators and irreducible lattices (2000, background): “For the Banach space HD L p .G; H/, classical inequalities (see [ Cl ]) show that (3.1) holds with tD 2 1C”
On James and von Neumann–Jordan constants and sufficient conditions for the fixed point property (2006, background): “Clarkson in 1936 [3], relates to the fixed point property has been widely studied.”