Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. also provide two applications: a Lusin-type approximation ${\mathbb {H}}$-rectifiable sets and coarea formula that completes program started in [18].

We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in CarnotCarathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold.

It is shown that local epi-sub-Lipschitz continuity of the function-valued mapping associated with a perturbed optimization problem yields the local Lipschitz continuity of the inf-projections (= marginal functions, = infimal functions). The use of the theorem is illustrated by considering perturbed nonlinear optimization problems with linear constraints.