Helly's Theorem and Centre Point

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When we say that the centre is nicely situatedin the middle, we actually mean that every linepassing through it roughly divides the set inhalf.The same result will be had if the points aredistributed uniformly in a circle. Anything thathas to do with circle, we hope to have a goodcentre.The word balanced is not a technical term. Iuse it very loosely because we will see laterthat we will need to stretch the meaning ofthis term quite a bit.12 min 7-1
What do We mean by Balanced<strong>Centre</strong>Every line through centre divides the set inhalf.So, is it possible to find a balanced centre inall cases?8
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 and 8: Various Notions of Centre• Like c
Page 9 and 10: Property We are Looking for inCentr
Page 11 and 12: Median is Centre in 1D1. We are not
Page 13: Centre of Uniformly Distributedpoin
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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