On stability of a class of convex functions
by
Mikhail V. Korobkov
Sobolev Institute of Mathematics, Novosibirsk, Russia

By the classical Hyers-Ulam stability theorem [1], if a function f:D Ì Rⁿ®R satisfies the inequalities f(tx+(1-t)y) £ tf(x)+(1-t)f(y)+e, 0 £ t £ 1, then there exists a convex function g:D®R such that sup_{x Î D}|f(x)-g(x)| £ K_ne. On the other hand, if f is locally approximated by convex functions (over small neighborhoods of points x Î D), then in general f need not be globally close to convex functions in the C-norm. For example, every differentiable function f can be locally approximated with high accuracy by linear functions but f could be rather nonconvex.

In this talk we discuss a certain stability result of such kind (local proximity Þ global proximity). It is natural to use Kopylov's notion of w-stability for classes of Lipschitz functions [2]. We recall that a class G of Lipschitz functions is called w-stable [2] if there exists a function s:[0, +¥)®[0, +¥) such that (1) s(e)®s(0)=0 as e®0; (2) the inequality w(f, G) £ s(W(f, G)) holds for every function f:D®R of a domain D Ì Rⁿ such that W(f, G) < ¥.

Here w(f, G)=sup_{B Ì D}w_B(f, G), W(f, G)=sup_{x Î D}{[`lim]_r®0w_{B(x, r)}(f, G)}, where B=B(x, r) is a ball in D and

wB(f, G) = inf g:B®R,  g Î G  {r-1 sup y Î B  |f(y)-g(y)|}.

The functionals w(·, G) and W(·, G) are referred to as the functionals of global and local proximity to the class G.

Theorem. Let G Ì Rⁿ be a compact set. Suppose that the projection of G on each straight line l Ì Rⁿ is a totally disconnected set. Then the class G of convex functions g satisfying g¢(x) Î G a.e. is w-stable.

References

1. Hyers D. H., Ulam S. M. Approximately convex functions // Proc. Amer. Math. Soc., 3, p. 821-828 (1952).

2. Kopylov A. P. On stability of isometric mappings // Sibirsk. Mat. Zh., 25, p. 132-144 (1984).

Date received: June 7, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-93.