A compact set of disjoint line segments in E 3 whose end set has positive measure

Mathematika ◽  
1971 ◽  
Vol 18 (1) ◽  
pp. 112-125 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Positive Measure ◽  
Compact Set ◽  
Line Segments ◽  
Disjoint Line ◽  
End Set

OPTIMAL BINARY SPACE PARTITIONS FOR SEGMENTS IN THE PLANE

2012 ◽  
Vol 22 (03) ◽  
pp. 187-205 ◽  
Author(s):  
MARK DE BERG ◽  
AMIRALI KHOSRAVI
Keyword(s):  
Recursive Partitioning ◽  
Np Hard ◽  
Line Segments ◽  
Minimum Number ◽  
Disjoint Line

An optimal BSP for a set S of disjoint line segments in the plane is a BSP for S that produces the minimum number of cuts. We study optimal BSPs for three classes of BSPs, which differ in the splitting lines that can be used when partitioning a set of fragments in the recursive partitioning process: free BSPs can use any splitting line, restricted BSPs can only use splitting lines through pairs of fragment endpoints, and auto-partitions can only use splitting lines containing a fragment. We obtain the following two results: • It is NP-hard to decide whether a given set of segments admits an auto-partition that does not make any cuts. • An optimal restricted BSP makes at most 2 times as many cuts as an optimal free BSP for the same set of segments.

scholarly journals Every Set of Disjoint Line Segments Admits a Binary Tree

2001 ◽  
Vol 26 (3) ◽  
pp. 387-410 ◽  
Author(s):  
P. Bose ◽  
M. E. Houle ◽  
G. T. Toussaint
Keyword(s):  
Binary Tree ◽  
Line Segments ◽  
Disjoint Line

scholarly journals On the Balayage for Logarithmic Potentials

1967 ◽  
Vol 29 ◽  
pp. 229-241 ◽  
Author(s):  
Nobuyuki Ninomiya
Keyword(s):  
Continuous Function ◽  
Compact Support ◽  
Positive Measure ◽  
Logarithmic Potential ◽  
Compact Set ◽  
Logarithmic Potentials ◽  
Capacity Zero

In this paper, we shall consider the logarithmic potential where μ is a positive measure in the plane, P and Q are any points and PQ denotes the distance from P to Q. In general, consider the potential of a positive measure μ taken with respect to a kernel K(P, Q) which is a continuous function in P and Q and may be + ∞ for P = Q. A kernel K (P, Q) is said to satisfy the balayage principle if, given any compact set F and any positive measure μ with compact support, there exists a positive measure μ′ supported by F such that K(P, μ′) = K(P, μ) on F with a possible exception of a set of k-capacity zero and K(P, μ′)≦K(P, μ) everywhere. A kernel K(P, Q) is said to satisfy the equilibrium principle if, given any compact set F there exists a positive measure λ supported by F such that K(P, λ) = V (a constant) on F with a possible exception of a set of K-capacity zero and K(p, λ)≦V everywhere.

scholarly journals Alternating paths through disjoint line segments

2003 ◽  
Vol 87 (6) ◽  
pp. 287-294 ◽  
Author(s):  
Michael Hoffmann ◽  
Csaba D Tóth
Keyword(s):  
Line Segments ◽  
Disjoint Line

scholarly journals Maximum Principles in the Potential Theory

1963 ◽  
Vol 23 ◽  
pp. 165-187 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Positive Measure ◽  
Hausdorff Space ◽  
Finite Energy ◽  
Maximum Principles ◽  
Compact Set ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Locally Compact Hausdorff Space

Ninomiya, in his thesis [13] on the potential theory with respect to a positive symmetric continuous kernel G on a locally compact Hausdorff space Ω, proves that G satisfies the balayage (resp. equilibrium) principle if and only if G satisfies the domination (resp. maximum) principle. He starts from the Gauss-Ninomiya variation and shows that for any given compact set K in Ω and any positive upper semi-continuous function u on K, there exists a positive measure μ on K such that its potential Gμ is ≥ u on the support of μ and Gμ≥u on K almost everywhere with respect to any positive measure with finite energy.

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