NDLI: Tropical Fermat--Weber Points

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lin, Bo Yoshida, Ruriko
Copyright Year	2018
Abstract	In a metric space, the Fermat--Weber points of a sample are statistics to measure the central tendency of the sample and it is well known that the Fermat--Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat--Weber points under the tropical metric on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {1}$ with a fixed $n \in \mathbb{N}$, motivated by its application to the space of equidistant phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat--Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for a key value associated with this set. We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat--Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat--Weber points.
Sponsorship	Coleman Foundation
Starting Page	1229
Ending Page	1245
Page Count	17
File Format	PDF
ISSN	08954801
DOI	10.1137/16M1071122
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-06-05
Access Restriction	Subscribed
Subject Keyword	ultrametrics $n$-dimensional polytopes General biology and biomathematics convex polytopes Applications of commutative algebra Fermat--Weber points tropical geometry
Content Type	Text
Resource Type	Article
Subject	Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Tropical Fermat-Weber points

Projections of Tropical Fermat-Weber Points

Moduli spaces of Delzant polytopes and symplectic toric manifolds

Projecting lattice polytopes without interior lattice points

On odd points and the volume of lattice polyhedra

Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling

Splitting Polytopes

Convex-normal (pairs of) polytopes

Three-dimensional lattice polytopes with two interior lattice points

Tropical Fermat--Weber Points

Similar Documents

Tropical Fermat-Weber points

Projections of Tropical Fermat-Weber Points

Moduli spaces of Delzant polytopes and symplectic toric manifolds

Projecting lattice polytopes without interior lattice points

On odd points and the volume of lattice polyhedra

Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling

Splitting Polytopes

Convex-normal (pairs of) polytopes

Three-dimensional lattice polytopes with two interior lattice points

Tropical Fermat--Weber Points