Stability in multidimensional Size Theory

TL;DR

This paper demonstrates that multidimensional size functions can be analyzed through a reduction to one-dimensional cases using a foliation approach, enabling new stability results and a novel distance measure.

Contribution

It introduces a method to reduce multidimensional size functions to 1D, defines a new distance, and proves their stability under this metric.

Findings

Reduction of multidimensional size functions to 1D cases

Definition of a new distance between size functions

Proof of stability of size functions with respect to this distance

Abstract

This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance.