Extremal radii, diameter and minimum width in generalized Minkowski spaces

AbstractWe introduce successive radii in generalized Minkowski spaces (that is, with respect to gauges) and study some first properties. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems, we also look at ball intersections and ball hulls.

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Semi-indefinite inner product and generalized Minkowski spaces

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2010.04.006 ◽

2010 ◽

Vol 60 (9) ◽

pp. 1190-1208 ◽

Cited By ~ 16

Author(s):

Á.G. Horváth

Keyword(s):

Inner Product ◽

Indefinite Inner Product ◽

Minkowski Spaces ◽

Generalized Minkowski Spaces

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The Restricted Arc-Width of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/1734 ◽

2003 ◽

Vol 10 (1) ◽

Author(s):

David Arthur

Keyword(s):

Unit Circle ◽

Single Point ◽

Minimum Width ◽

Graph Operations ◽

Function Mapping

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.

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R-DFS: A Coverage Path Planning Approach Based on Region Optimal Decomposition

Remote Sensing ◽

10.3390/rs13081525 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1525

Author(s):

Gang Tang ◽

Congqiang Tang ◽

Hao Zhou ◽

Christophe Claramunt ◽

Shaoyang Men

Keyword(s):

Path Planning ◽

Google Earth ◽

Simulation Experiments ◽

Minimum Width ◽

Coverage Path Planning ◽

Optimal Paths ◽

Planning Approach ◽

Polygonal Area ◽

Aerial Vehicle ◽

Working Distance

Most Coverage Path Planning (CPP) strategies based on the minimum width of a concave polygonal area are very likely to generate non-optimal paths with many turns. This paper introduces a CPP method based on a Region Optimal Decomposition (ROD) that overcomes this limitation when applied to the path planning of an Unmanned Aerial Vehicle (UAV) in a port environment. The principle of the approach is to first apply a ROD to a Google Earth image of a port and combining the resulting sub-regions by an improved Depth-First-Search (DFS) algorithm. Finally, a genetic algorithm determines the traversal order of all sub-regions. The simulation experiments show that the combination of ROD and improved DFS algorithm can reduce the number of turns by 4.34%, increase the coverage rate by more than 10%, and shorten the non-working distance by about 29.91%. Overall, the whole approach provides a sound solution for the CPP and operations of UAVs in port environments.

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