  Convex Integration is a theory developed in the '70s by M. Gromov. This
theory allows to solve families of differential problems satisfying some convex
assumptions. From a subsolution, the theory iteratively builds a solution by
applying a series of convex integrations. In a previous paper arXiv:1909.04908,
we proposed to replace the usual convex integration formula by a new one called
Corrugation Process. This new formula is of particular interest when the
differential problem under consideration has the property of being of Kuiper.
In this paper, we consider the differential problem of $\epsilon$-isometric
maps and we prove that it is Kuiper in codimension 1. As an application, we
construct $\epsilon$-isometric maps from a short map having a conical
singularity.
