Stability of affine approximations on bounded domains

Abstract

Let f be an arbitrary function defined on a convex subset G of a linear space. Suppose the restriction of f on every straight line can be approximated by an affine function on that line with a given precision ε > 0 (in the uniform metric); what is the precision of approximation of f by affine functionals globally on G? This problem can be considered in the framework of stability of linear and affine maps. We show that the precision of the global affine approximation does not exceed C(log d)ε, where d is the dimension of G, and C is an absolute constant. This upper bound is sharp. For some bounded domains G ⊂ ℝd, it can be improved. In particular, for the Euclidean balls the upper bound does not depend on the dimension, and the same holds for some other domains. As auxiliary results we derive estimates of the multivariate affine approximation on arbitrary domains and characterize the best affine approximations.