Strong unicity versus modulus of convexity

Abstract

We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p-type modulus of convexity implies order p strong unicity of the simultaneous approximant.The study of strong unicity, initiated by Newman and Shapiro [9], resulted from a concern with the conditioning of the best approximation problem. In the present paper, the relation between strong unicity and modulus of convexity is explored. Throughout our discussion, we suppose that (X, ||-||) is a Banach space with modulus of convexity 8 > 0, and that V is a closed subspace of X. Bjornestal [2] showed that if x 0 G S(X) and if ||zo -2/|| ^ ll z o|| for every y G V, then, for every y G V with ||y|| ^ 3, that is, the vector 0 is a strongly unique best approximation to xo from V. We shall now show that Bjornestal's inequality characterises the modulus of convexity in a certain context, and that an analogue of this inequality holds in the non-uniformly convex space which underlies the theory of simultaneous approximation. We hope that these results will contribute to a deeper understanding of uniform convexity, as well as to the more "practical" goal of finding conditions under which tlie best approximation operator is continuous.We begin with precise definitions of the relevant concepts. Suppose that f £ X is fixed. An element z G V is said to be a best \\-\\-approximant to / from V if \\z -f\\ = inf{||w -f\\ : w G V}. Since X is uniformly convex, there is exactly one such z, and we denote it by /*. We say that /* is (M,j)-strongly unique of order p if, for some M -M(f) > 0, there exists a 7 = y(f,M) > 0 such that, for all v G V with | | » -/ ' | | < M

Nonlinear Chebyshev approximation to set-valued functions

“…The next result extends ( [19], Theorem 5) for Chebyshev approximation and a class G more general than a closed subspace. Theorem 4.4: Let G ⊂ C(X) be a family with the weak betweeness property, and f 1 , f 2 ∈ C(X).…”

Proof Mining: A Systematic Way of Analysing Proofs in Mathematics

“…Among the many publications on the connection between moduli of uniform convexity and rates of strong unicity see e.g. [14,35,66,77].…”

Continuities of Metric Projection and GeometricConsequences

“…Note that 6 K (x) is the convex set of minima of the function F x restricted to K. We say that 6 K (x) is a set of weak sharp minima for F x relative to K if there is an :>0 such that &x& y& dist(x, K )+: } dist( y, 6 K (x)) for every y # K (cf. [12,13,16,18,20,28,29,30,31,32,33] for some related research on strong uniqueness and weak sharp minima). We say that the metric projection 6 K has the weak sharp minimum property if 6 K (x) is a set of weak sharp minima for every x # R n .…”

“…The metric projection onto every subspace of a finitedimensional normed linear space X is continuous if and only if the unit ball of X is totally tubular. Huotari and Sahab [20] showed that in certain cases the modulus of convexity of the norm is characterized in terms of the order of strong unicity of the metric projection. All these results show that there is a connection between the various continuity conditions of metric projections and the geometric characteristics (or the``shape'') of the unit ball in a normed linear space.…”