Stability and computation in multidimensional size theory
Andrea Cerri · AMS Dottorato Institutional Doctoral Theses Repository (University of Bologna) · 2007
P ∈M ϕ(P )ψ(f (P )) ∞ , can easily prove the stability of this new descriptor (and hence of the corresponding k-dimensional size function) also in higher dimensions (Proposition 2.2. 2 and Proposition 2.2.3), by using a recent result of stability proved for 1dimensional size functions (cf.[17,18]).Moreover, we show that a matching distance between size functions, with respect to measuring functions taking values in R k , can easily be introduced (Definition 2.2), providing a lower bound for the natural pseudo-distance (Theorem 2.2.4).All these results open the way to the use of Multidimensional Size Theory in real applications.Outline.This thesis contains the results of our research activity carried out in the last years in order to face the problem of the extension to the multidimensional case of Size Theory (cf.[12,13]).In Chapter 1 we give preliminary definitions for k-dimensional size functions, and the main results for the particular case k = 1.In Chapter 2 the foliation we use is presented, and the reduction to the 1-dimensional case is proved.Moreover, we show the stability of our computational method, implying a lower bound for the natural pseudo-distance.Additionally, a new distance between multidimensional size functions is introduced.Chapter 3 is devoted to present some computational techniques for computing k-dimensional size functions in applications.Some discussion and proposals for future research conclude the thesis.