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Total diameter and area of closed submanifolds [article]

Mohammad Ghomi, Ralph Howard
2015 arXiv   pre-print
The total diameter of a closed planar curve C⊂ R^2 is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C.  ...  Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle.  ...  We also thank Igor Belegradek for locating the reference [21] . Finally, thanks to the anonymous referee for suggesting improvements to an earlier draft of this work.  ... 
arXiv:1312.1384v2 fatcat:amu7kettgfazrdisyeljpft6tm

The chord length distribution function of a non-convex hexagon

Uwe Bäsel, Vittoria Bonanzinga, Andrei Duma
2018 Communications in Applied and Industrial Mathematics  
In this paper we obtain the chord length distribution function of a non-convex equilateral hexagon and then derive the associated density function.  ...  Finally, we calculate the expected value of the chord length.  ...  The chord length distribution functions for a number of planar convex figures are already known.  ... 
doi:10.1515/caim-2018-0002 fatcat:if7acxvzfra3xofjaoz3sbk5rm

Page 1589 of Mathematical Reviews Vol. , Issue 86d [page]

1986 Mathematical Reviews  
Voss, Klaus (DDR-HUMB) 86d:52001 Integrals of chord length powers for planar convex figures. (German and Russian summaries) Elektron. Informationsverarb. Kybernet. 20 (1984), no. 7-9, 488-494.  ...  Let s(g) be the length of the chord Fg, where g is a line and F a convex body, both in the Euclidean plane.  ... 

Circles Minimize most Knot Energies [article]

Aaron Abrams, Jason Cantarella, Joseph H. G. Fu, Mohammad Ghomi, and Ralph Howard
2001 arXiv   pre-print
The proof is based on a theorem of G. Luko on average chord lengths of closed curves.  ...  Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle.  ...  We thank Kostya Oskolkov for pointing out that the complex form of Fourier series would simplify the first proof of Theorem 2.2.  ... 
arXiv:math/0105138v1 fatcat:wzaapjcbfvdqpd4vkpcyoxoofa

Page 1492 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews  
The chord powers S,, (i.e., perimeter length times nth moment of an invariant random chord) are evaluated for n=5, 6, 7 and 8 for certain symmetric planar convex figures via the principal moments of inertia  ...  [Voss, Klaus] Powers of chords for convex sets. (German summary) Biometrical J. 2A (1982), no. 5, 513-516.  ... 


Magdalena Hykšová, Anna Kalousová, †Ivan Saxl
2012 Image Analysis and Stereology  
The paper provides an account of the history of geometric probability and stereology from the time of Newton to the early 20th century.  ...  On the other hand, practical rules of the estimation of area or volume fraction and other characteristics, easily deducible from geometric probability theory, were proposed without the knowledge of this  ...  ACKNOWLEDGEMENTS The authors acknowledge the financial support for this work provided by GAAV, grant IAA801240901, and by the grant MSM 0021620839.  ... 
doi:10.5566/ias.v31.p1-16 fatcat:5pxthnj7xfbdpbaswuzmmjpnti

Characterization of balls by generalized Riesz energy [article]

Jun O'Hara
2017 arXiv   pre-print
is defined as the double integral of some power of the distance between pairs of points.  ...  We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy  ...  In fact, for convex bodies, Mallows and Clark [MC] gave a pair of non-congruent convex planar polygons with the same chord length distribution as illustrated in Figure 1 Waksman [W] pointed out that  ... 
arXiv:1707.02405v1 fatcat:jlwrwtq63bhfreajr5ws3n7qzu

Gauss equation and injectivity radii for subspaces in spaces of curvature bounded above [article]

Stephanie B. Alexander, Richard L.Bishop
2005 arXiv   pre-print
A Gauss equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K.  ...  That is, a subspace of extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature less than or equal to K+A^2.  ...  Acknowledgments Our interest in a Gauss Equation in Alexandrov spaces of curvature bounded above stems from discussions with David Berg and Igor Nikolaev in the mid-90's, for which we thank them.  ... 
arXiv:math/0511570v1 fatcat:h4xf2usrarhcfdn3okethfm5bm

The elastica problem under area constraint [article]

Vincenzo Ferone, Bernd Kawohl, Carlo Nitsch
2015 arXiv   pre-print
The proof is of a geometric nature and deforms parts of γ in a finite number of steps to construct some related convex sets with smaller energy.  ...  We show that the elastic energy E(γ) of a closed curve γ has a minimizer among all plane simple regular closed curves of given enclosed area A(γ), and that the minimum is attained for a circle.  ...  We thank A.Henrot for helpful discussions during a visit to Napoli in 2013.  ... 
arXiv:1411.6100v2 fatcat:2ga6qvbrw5aq3js62bmjg5wtfa

Lower and upper bounds for chord power integrals of ellipsoids

Lothar Heinrich
2014 Applied Mathematical Sciences  
First we discuss different representations of chord power integrals I p (K) of any order p ≥ 0 for convex bodies K ⊂ R d with inner points.  ...  A further inequality conjectured in Davy (1984) is proved for ellipsoids. Some remarks on chord power integrals of superellipsoids and simplices round off the topic.  ...  Chord Power Integrals -Definition and Basics Let K be a convex body in R d with interior points and S d−1 = ∂B d the boundary of the Euclidean unit ball B d = {x ∈ R d : x ≤ 1}.  ... 
doi:10.12988/ams.2014.411913 fatcat:lbjf4ah2mvahzahswh46g55zd4

On the geometric dilation of closed curves, graphs, and point sets

Adrian Dumitrescu, Annette Ebbers-Baumann, Ansgar Grüne, Rolf Klein, Günter Rote
2007 Computational geometry  
The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H.  ...  The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean  ...  Acknowledgement We would like to thank John Sullivan and Salvador Segura-Gomis for helpful discussions and the anonymous referees for their valuable comments.  ... 
doi:10.1016/j.comgeo.2005.07.004 fatcat:u3hewviktbf7nlsmna435jq4qu

Evaluation of integral forces and pressure fields from planar velocimetry data for incompressible and compressible flows

B. W. van Oudheusden, F. Scarano, E. W. M. Roosenboom, E. W. F. Casimiri, L. J. Souverein
2007 Experiments in Fluids  
The approach to determine pressure fields and integral loads from planar velocimetry data is discussed, in relation to the implementation for incompressible and compressible flows around twodimensional  ...  The second topic considers the extension of the method to steady compressible flow, with the supersonic flow around a bi-convex airfoil as experimental test case.  ...  Tests were carried out on a full span bi-convex airfoil with a chord of 100 mm and a thickness of 12 mm.  ... 
doi:10.1007/s00348-007-0261-y fatcat:w5tkgbi74ncnjdxxytydhly5sy

A Survey of Shape Feature Extraction Techniques [chapter]

Yang Mingqiang, Kpalma Kidiyo, Ronsin Joseph
2008 Pattern Recognition Techniques, Technology and Applications  
Chord distribution The basic idea of chord distribution is to calculate the lengths of all chords in the shape (all pair-wise distances betwee boundary points) and to build a histogram of their lengths  ...  Chord length function The chord length function is derived from shape boundary without using any reference point.  ...  This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided  ... 
doi:10.5772/6237 fatcat:ggyqwzt4jvfuxg4cjb7vjniu3q

Conceptual Design and Performance Optimization of a Tip Device for a Regional Turboprop Aircraft

Ilias Lappas, Akira Ikenaga
2019 Aerospace (Basel)  
The installation of wing tip devices has not been a popular choice for regional turboprop aircraft, and the novelty of the current study is to investigate the feasibility of retrofitting the British Aerospace  ...  Since successful winglet retrofit programs for typical short to medium-range narrow-body aircraft report even more than 3% of block fuel improvements, undertaking the project of installing an optimal winglet  ...  Acknowledgments: The authors would like to thank Luuk van der Schaft, Odeh Dababneh, and Vishagen Ramasamy for contributing with ideas and advice.  ... 
doi:10.3390/aerospace6100107 fatcat:k5pm5oe6iva7zms3w5gjd5hgwq

Introducing symplectic billiards [article]

Peter Albers, Serge Tabachnikov
2017 arXiv   pre-print
As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function.  ...  We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.  ...  It remains to use the Lusternik-Schnirelman lower bound for the number of critical points given by the category of the lens space L.  ... 
arXiv:1708.07395v1 fatcat:eu2q5t3znjdmpgakbq5qpzr3o4
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