Helly's Theorem and Centre Point

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Like there is a central place in China whichis farthest from the sea in any direction. Itsdistance to Indian Ocean, Pacific Ocean <strong>and</strong>Arctic Ocean is same.In 1-dimension both the notions generalise tothe mid-point of the two exterior point. Hencethese notions are not at all suitable. Even averageof n values captures more informationthan these.By the way, we usually use average because it iseasier to compute, but one or two very extremevalues can have adverse effect on average.5 min 4-1
Property We are Looking for in<strong>Centre</strong>• If possible, equal count/area/volume on allsides• Otherwise, balancing these as much as possible5
Page 1 and 2: Helly’s Theorem and CentrePointSw
Page 3 and 4: Helly’s theorem (1923) and Radon
Page 5 and 6: Motivation for Centre Points• We
Page 7: Various Notions of Centre• Like c
Page 11 and 12: Median is Centre in 1D1. We are not
Page 13 and 14: Centre of Uniformly Distributedpoin
Page 15 and 16: What do We mean by BalancedCentreEv
Page 17 and 18: Worst Case Distribution in 2D• Me
Page 19 and 20: Worst Case Distribution in 3DCentre
Page 21 and 22: Definition of a Centre-pointA centr
Page 23 and 24: Remarks on the Definition ofCentre-
Page 25 and 26: Definition of a Centre-point inLowe
Page 27 and 28: Questions We can Raise• Do centre
Page 29 and 30: The Answers• Do centre-points alw
Page 31 and 32: Task Before UsHow do we prove the e
Page 33 and 34: Helly’s TheoremWe need to take he
Page 35 and 36: Bound d + 1 is tight - I• Counter
Page 37 and 38: Bound d + 1 is tight - II• Same E
Page 39 and 40: Bound d + 1 is tight (SimpleCase) -
Page 41 and 42: Proof of Helly’s Theorem• We ne
Page 43 and 44: Radon’s TheoremRadon’s Theorem
Page 45 and 46: What is the Meaning ofRadon’s The
Page 47 and 48: Sketch of the Proof of Radon’sThe
Page 49 and 50: Explanation of ProofCase of five po
Page 51 and 52: Proof of Helly’s Theorem• The p
Page 53 and 54: Explanation of ProofCase of four se
Page 55 and 56: Observation One forCentre-points•
Page 57 and 58: Observation two forCentre-points•
Page 59 and 60:
Proof of Centre Point• Every d+1
Page 61 and 62:
Computation of Centre-Point• Cent
Page 63 and 64:
Computation of Centre-Point inPlane
Page 65 and 66:
Computation of ApproximationCentre-
Page 67 and 68:
Checking Centre-Point• Problem of
Page 69 and 70:
A Generalisation of Helly’sTheore
Page 71 and 72:
Tverberg’s Theorem — AGeneralis
Page 73 and 74:
Summary• We saw the Helly’s The
Page 75:
Thank You39
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Helly's Theorem and Centre Point

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