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Minimum Perimeter-Sum Partitions in the Plane
[article]

2017
*
arXiv
*
pre-print

We consider

arXiv:1703.05549v1
fatcat:kahih4brcnhmhowtnki3jeseky
*the*problem of*partitioning*P into two subsets P_1 and P_2 such that*the**sum*of*the**perimeters*of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes*the*convex hull of P_i. ... Let P be a set of n points*in**the**plane*. ... This research was initiated when*the*first author visited*the*Department of Computer Science at TU Eindhoven during*the*winter 2015-2016. ...##
###
Minimum Perimeter-Sum Partitions in the Plane

2019
*
Discrete & Computational Geometry
*

We consider

doi:10.1007/s00454-019-00059-0
fatcat:le5hvb7afzhtjeqwxieegcaatm
*the*problem of*partitioning*P into two subsets P 1 and P 2 such that*the**sum*of*the**perimeters*of ch(P 1 ) and ch(P 2 ) is minimized, where ch(P i ) denotes*the*convex hull of P i . ... Let P be a set of n points*in**the**plane*. ... This research was initiated when*the*first author visited*the*Department of Computer Science at TU Eindhoven during*the*winter 2015-2016. ...##
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Squarepants in a tree

2009
*
ACM Transactions on Algorithms
*

within each cluster, and

doi:10.1145/1541885.1541890
fatcat:aiwsaw6ajveenmiohzwbxfr2ii
*in**the*hyperbolic or Euclidean*planes*, minimizing*the**sum*of cluster*perimeters*. ... Our algorithms for*the*hyperbolic and Euclidean*planes*can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves*partitioning**the**plane*minus*the*input points into ... Approximate clustering for hyperbolic*planes*If a point set has all points at least constant distance apart Then convex hull*perimeter*≈*minimum*spanning tree length Proof idea: Show each of*the*following ...##
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Tiling a Plane in a Dynamical Process and its Applications to Arrays of Quantum Dots, Drums, and Heat Transfer

2005
*
Physical Review Letters
*

Thus,

doi:10.1103/physrevlett.95.088304
pmid:16196909
fatcat:i2mv23khfvgarktfwjuko6un7a
*the*resulting*partition*of*the**plane*is determined by minimization of*the**sum*of*the*eigenvalues, and not by*the*minimization of*the*total*perimeter*of*the*cells as*in**the*famous honeycomb problem ...*The*evolution of*the*system leads to*the**partition*of*the**plane*into cells, each occupied by only one component. For large N,*the*stationary state becomes a periodic array of hexagonal cells. ... Thus,*the*resulting*partition*of*the**plane*is determined by minimization of*the**sum*of*the*eigenvalues, and not by*the*minimization of*the*total*perimeter*of*the*cells as*in**the*famous honeycomb problem ...##
###
Convex Regions and their 'Fairest' Equipartitioning Fans
[article]

2012
*
arXiv
*
pre-print

We discuss

arXiv:1208.6508v1
fatcat:dgkfrokpvrhhbm2pjkymj67qke
*the**partition*of convex 2D regions into n (a positive integer) equal area convex pieces by fans with*the*following additional requirement:*the**perimeters*of*the*resultant equal area pieces should ... A k-fan is a set of k half-lines (rays) all starting from*the*same point, called*the*origin of*the*fan. ... Let us count*the*changes*in*sign of*the*change*in**perimeter*of each piece during this rotation. It can be seen that*the**sum*over all n pieces of this count of 'switches' would be only O(m). ...##
###
Page 3044 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

If a polyomino or a pseudo-polyomino is constructed

*in*this lattice, then*the*associated value is*the**sum*of assigned numbers*in*its cells. ... It is shown that, for skeletons L, this*partition*yields direct*sum*decompositions of*the*submodular polytope associated with f. An example of such an L is given by*the*collection of minimizers of /. ...##
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The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

2010
*
Electronic Journal of Combinatorics
*

Candidates to

doi:10.37236/317
fatcat:hxwterqfwzgg7drxhvv6bdt6xy
*the*least*perimeter**partition*of various polygonal shapes into $N$ planar connected equal-area regions are calculated for $N\le 42$, compared to*partitions*of*the*disc, and discussed*in**the*...*Partitions*of a square and a pentagon show greater disorder. Candidates to*the*least*perimeter**partition*of*the*surface of*the*sphere into $N$ connected equal-area regions are also calculated. ... Table 2 :*Perimeter*E/R and topology of*the*candidates to*the**minimum**perimeter**partition*into equal-area bubbles of*the*surface of*the*unit sphere. ...##
###
THE TRAVELING SALESMAN PROBLEM FOR LINES AND RAYS IN THE PLANE

2012
*
Discrete Mathematics, Algorithms and Applications (DMAA)
*

*In*

*the*path variant, we seek a shortest path that visits each region. ...

*In*

*the*Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. ... Let now Q * be an intersecting rectangle of L with

*minimum*

*sum*of

*the*lengths of three sides. ...

##
###
On the honeycomb conjecture and the Kepler problem
[article]

2009
*
arXiv
*
pre-print

First, we proved that

arXiv:0906.1249v4
fatcat:a5rkrghn65fdrbjivjly5yfrfi
*the*regular hexagons are*the*only 2-dim blocks that have unit area and*the*least*perimeter*(or contain a unit circle and have*the*least area) that tile*the**plane*. ... Finally,*the*Kepler conjecture can also be proved to be true by introducing*the*concept of*the**minimum*2-dim and 3-dim Kepler building blocks. ...*The*central point*in**the*honeycomb conjecture is therefore to determine those suitable regular polygons of unit area and least*perimeter*that will tile*the*whole*plane*without leaving gaps or overlapping ...##
###
CONTENTS
[chapter]

1954
*
Mathematics and Plausible Reasoning, Volume 1
*

Triangle with

doi:10.1515/9780691218304-toc
fatcat:5ioz6wgwzvf6vhepm37zku2uty
*minimum**perimeter*inscribed*in*a given triangle. 9. Traffic center of four points*in*space. 10. Traffic center of four points*in*a*plane*. 11. ...*Minimum*and maximum distances*in**plane*geometry. 2.*Minimum*and maximum distances*in*solid geometry. 3. Level lines*in*a*plane*. 4. Level surfaces*in*space. 11. ...##
###
The Least-Perimeter Partition of a Sphere into Four Equal Areas

2009
*
Discrete & Computational Geometry
*

We prove that

doi:10.1007/s00454-009-9197-8
fatcat:44oxtwllsbctnfatfu6rndwzke
*the*least-*perimeter**partition*of*the*sphere into four regions of equal area is a tetrahedral*partition*. ...*In*a*perimeter*-minimizing*partition*each region has a pressure, defined up to addition of a constant, so that*the*difference*in*pressures between regions A and B is*the**sum*of*the*(signed) curvatures crossed ...*plane*into unit areas. ...##
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The traveling salesman problem for lines and rays in the plane
[article]

2012
*
arXiv
*
pre-print

Along

arXiv:1204.5828v1
fatcat:2r45nl25drbcjd6j25uj7wddh4
*the*way we derive a tight bound on*the**minimum**perimeter*of a rectangle enclosing an open curve of length L. ...*In**the*path variant, we seek a shortest path that visits each region. ... It can be verified that*the**perimeter*of any other enclosing rectangle is larger (details*in**the*Appendix), hence*the*rectangle Q enclosing γ constructed*in*our proof has*minimum**perimeter*. ...##
###
Page 7255 of Mathematical Reviews Vol. , Issue 2001J
[page]

2001
*
Mathematical Reviews
*

Under reasonable computa- tional models, we obtain lower bounds on

*the**minimum*amount of work required to maintain any binary space*partition*of mov- ing segments*in**the**plane*or any Steiner triangulation ... of moving points*in**the**plane*. ...##
###
Page 6369 of Mathematical Reviews Vol. , Issue 87k
[page]

1987
*
Mathematical Reviews
*

There are some well-known elementary geometric statements concerning

*the*points, triangles, polygons, etc.,*in**the*Euclidean*plane*, which are proved also*in**the*hyperbolic*plane*if*the*points involved ...*The*problem is then to fill*in**the*matrix of interpoint distances with a*minimum*error based on some norm. They propose using*the*distances of*the*points from same reference object as coordinates. ...##
###
Least perimeter partition of the disc into N bubbles of two different areas

2019
*
The European Physical Journal E : Soft matter
*

When

doi:10.1140/epje/i2019-11857-0
fatcat:uq4jzm2zd5dqbpzuhvrqpommjq
*the*area ratio is significantly far from one,*the*least*perimeter**partitions*tend to have a "mixed" configuration,*in*which bubbles of*the*same area are not adjacent to each other. ... We present conjectured candidates for*the*least*perimeter**partition*of a disc into N ≤ 10 connected regions which take one of two possible areas. ... , and reproduction*in*any medium, provided*the*original work is properly cited. ...
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