Abstract
This short chapter proves the equivalence of the three mathematical models generalizing the Gilbert-Steiner problem to continuous measures. The proof of this equivalence involves most structure properties proved in Chapter 7. Section 9.1 tells us in short that the Gilbert-Steiner and the optimal traffic plan problems are equivalent for finite atomic measures. Section 9.2 is dedicated to the equivalence of the traffic plan model with the Maddalena et al. irrigation pattern model [59]. Finally Section 9.3 treats the equivalence between traffic plans and Xia’s tranport paths. In all cases the equivalence is strong, as not only the minimal energies are the same but the minimizing objects, of very diverse nature, can be identified with each other. The presentation here follows [13]. Several tools come from [58] and [70].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2009). The Equivalence of Various Models. In: Optimal Transportation Networks. Lecture Notes in Mathematics, vol 1955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69315-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-69315-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69314-7
Online ISBN: 978-3-540-69315-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)