Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function $$f:X\rightarrow X$$ f : X → X is uniformly expansive on a set $$\varLambda \subset X$$ Λ ⊂ X if there exist $$\varepsilon >0$$ ε > 0 and $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) such that for any two orbits $$\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda $$ x : { - N , … , N } → Λ , $$\hbox {v}:\{-N,\ldots ,N\} \rightarrow X$$ v : { - N , … , N } → X of $$f$$ f we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ sup - N ≤ n ≤ N d ( x n , v n ) ≤ ε ⇒ d ( x 0 , v 0 ) ≤ α sup - N ≤ n ≤ N d ( x n , v n ) . It occurs that a function is uniformly expansive iff there exists a generalized cone-field on $$X$$ X such that $$f$$ f is cone-hyperbolic.