Helly's Theorem and Centre Point

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Equivalently, every hyper-plane passing throughthe centre-point contains at least ⌈ nd+1 ⌉ points.Mostly we will be looking at the discrete <strong>and</strong>finite set of points. So we will be interestedonly in the count. But it is possible to generalisefor other objects, if we try to balancearea, volume, etc.24 min 11-1
Remarks on the Definition of<strong>Centre</strong>-pointWe will concentrate only in E d , even thoughcentre-points exists elsewhere.What is important is the notion of half-space.Not all geometrical spaces may have this notion.12
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 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: Definition of a Centre-pointA centr
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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