Аннотация: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 \varepsilon > 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) \varepsilon, where d
is the dimension of G, and C is an absolute constant. This upper bound is sharp. For
some bounded domains G ⊂ R^d 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 estimations of the multivariate affine
approximation on arbitrary domains and characterize the best affine approximations.