On metric stability of set-valued subadditive functions

Abstract

In Shulman (J Funct Anal 263:1468–1484, Theorem 2.2) the author estimated the cardinality of the global image of a subadditive map from a group to an arbitrary power set and applied the result to functional equations of Levi-Civita type (the relations of this result to covering a group by subgroups will be shortly discussed in the present paper). Here we will establish the metric stability of this theorem. Namely it will be shown that if a set-valued map F from a group G to a metric space (mathcal {X}, d) satisfies the condition dF(gh),F(g)∪F(h)<δ,for anyg,h∈Gand someδ>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d \\left( F(gh), F(g)\\cup F(h)\\right) < \\delta , \\;\\; \ ext {for any }\\ g,h\\in G \\; \ ext { and some } \\delta >0, \\end{aligned}$$\\end{document}and the cardinality of each F(g) does not exceed nin mathbb {N}, then the set F(G):= cup _{gin G}F(g) can be covered by n(n+3)/2 balls of radius (3cdot 6^{n-1}-1)delta .