Continuity of extensions of Lipschitz maps and of monotone maps

Let X be a subset of a Hilbert space. We prove that if v X→ Rᵐ is such that {equation*} v(x)-∑ᵢ₌₁^m t_iv(x_i)≤ x-∑ᵢ₌₁^m t_ix_i {equation*} for all x,x₁,,xₘᵐ and all non-negative t₁,,tₘ that add up to one, then for any $1$-Lipschitz u Aᵐ, with A⊂ X, there exists a $1$-Lipschitz extension ũ Xᵐ of u such that the uniform distance on X between ũ and v is the same as the uniform distance on A between u and v. Morever, if either m∈ \1,2,3\ or X is convex, we prove the converse: we show that a map v Xᵐ that allows for a $1$-Lipschitz, uniform distance preserving extension of any $1$-Lipschitz map on a subset of X also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.