On metric stability of set-valued subadditive functions

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