We consider compact invariant sets \( \Lambda \) for \( C^{1} \) maps in arbitrary dimension. We prove that if \( \Lambda \) contains no critical points then there exists an invariant probability measure with a Lyapunov exponent \( \lambda \) which is the \emph{minimum} of all Lyapunov exponents for all invariant measures supported on \( \Lambda \). We apply this result to prove that \( \Lambda \) is \emph{uniformly expanding} if every invariant probability measure supported on \( \Lambda \) is hyperbolic repelling. This generalizes a well known theorem of Ma\~n\'e to the higher--dimensional setting.