Publication at Faculty of Mathematics and Physics |
2018
Abstract
We prove that for a normed linear space X, if f(1) : X -> R is continuous and semiconvex with modulus omega, f(2) : X -> R is continuous and semiconcave with modulus omega and f(1) <= f(2), then there exists f is an element of C-(1,omega)(X) such that f(1) <= f <= f(2). Using this result we prove a generalization of Ilmanen lemma (which deals with the case omega(t) = t) to the case of an arbitrary nontrivial modulus omega.
This generalization (where a C-loc(1,omega) function is inserted) gives a positive answer to a problem formulated by A. Fathi and M.
Zavidovique in 2010.