Four-point Fermat location problems revisited. New proofs and extensions of old results

Plastria, Frank

doi:10.1093/imaman/dpl007

Cited by 15 publications

(12 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…If we want to know a comparative between mean and median location, then we show you a quadrilateral, its diagonals, contour plots of the system Although we are aware that this extension is investigated in several complete studies in various articles, among them we will highlight that of Plastria (2006), which deals with the problem by using the classification of Kupitz and Martini (1997), describing the properties in the following way from the beginning:…”

Section: Preprintmentioning

confidence: 99%

Statistical Significance of the Median of a Set of Points on the Plane

Verdejo

Viveros

Uclés

2021

The College Mathematics Journal

View full text Add to dashboard Cite

Section: Preprintmentioning

confidence: 99%

Statistical Significance of the Median of a Set of Points on the Plane

Verdejo

Viveros

Uclés

2021

The College Mathematics Journal

View full text Add to dashboard Cite

“…i=0 make a convex quadrilateral then their FT point is located at the intersection of the diagonals of that quadrilateral; on the other hand, if one of the points is inside the triangle determined by the convex hull of the other three (i.e., it is a convex combination of the other three points), then the FT point is the point inside the triangle (see Ref. [44]). Using this property in Ref.…”

Section: Multiple Binary Qubit Povmsmentioning

confidence: 99%

Joint measurability structures realizable with qubit measurements: Incompatibility via marginal surgery

Nikola

Kunjwal

2020

Phys. Rev. Research

View full text Add to dashboard Cite

Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable. An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements. Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs. On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures. Here we explore the scope of joint measurability structures realizable with qubit POVMs. We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of compatible measurements. We show explicit examples of marginal surgery on a special set of qubit POVMs to construct joint measurability structures such as N-cycle and N-Specker scenarios for any integer N 3. We also show the realizability of various joint measurability structures with N ∈ {4, 5, 6} vertices. In particular, we show that all possible joint measurability structures with N = 4 vertices are realizable. We conjecture that all joint measurability structures are realizable with qubit POVMs. This contrasts with the unbounded dimension required in R. Kunjwal et al. [Phys. Rev. A 89, 052126 (2014)]. Our results also render this previous construction maximally efficient in terms of the required Hilbert space dimension. We also obtain a sufficient condition for the joint measurability of any set of binary qubit POVMs which powers many of our results and should be of independent interest.

show abstract

“…Also, it is well known that given four coplanar points, if one of those points lies inside the triangle formed by the other three points, then the Fermat point (or geometric median) is that point. Otherwise, the four points form a convex quadrilateral and the Fermat point is the intersection point of the diagonals of such quadrilateral (see [40]).…”

Section: Remarkmentioning

confidence: 99%