  In this paper, we prove a quantitative relative index theorem. It provides a
conceptual framework for studying some conjectures and open questions of Gromov
on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz
rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with
certain types of subsets removed. This $\lambda$-Lipschitz rigidity theorem is
asymptotically optimal. As a consequence, we obtain an asymptotically optimal
$\lambda$-Lipschitz rigidity theorem for positive scalar curvature metrics on
hemispheres. These give positive answers to the corresponding open questions
raised by Gromov. As another application, we prove Gromov's $\square^{n-m}$
inequality on the bound of distances between opposite faces of spin manifolds
with cube-like boundaries with a suboptimal constant. As immediate
consequences, this implies Gromov's cube inequality on the bound of widths of
Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian
bands with suboptimal constants. Further geometric applications will be
discussed in a forthcoming paper.
