Asymptotics of mean value points: a survey

Abstract

This expository article is devoted to some results about non-trivial asymptotical estimates of mean value points in analytical and geometrical theorems. In particular, we provide detailed proofs of the first and the second Ionin’s conjectures, respectively concerning with the first mean value theorem for integrals and the Taylor formula with the Lagrange remainder. We consider also some related results on divided differences and the multi-dimensional mean value theorem for integrals. In the final section, we deal with the asymptotic behavior of support points for planar continuous curves and tangent points for differentiable curves in Euclidean spaces.