# Preliminaries to Main Results

# Preliminaries to Main Results

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of *n*-dimensional surfaces. The idea is to deform a flat surface passing through a point *x* (along which we imagine that a certain function *f* is almost affine) to a surface passing through a point witnessing that *f* is not Fréchet differentiable at *x*. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the function*f* along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ωⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

*Keywords:*
regular differentiability, derivative, deformation, flat surface, linear operators, tensor products, Fréchet differentiability, Lipschitz map, Euclidean space

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