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| In [[integral geometry]] (otherwise called geometric probability theory), '''Hadwiger's theorem''' characterises the [[valuation (measure theory)|valuations]] on [[convex body|convex bodies]] in '''R'''<sup>''n''</sup>. It was proved by [[Hugo Hadwiger]].
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| ==Introduction==
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| ===Valuations===
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| Let '''K'''<sup>''n''</sup> be the collection of all compact convex sets in '''R'''<sup>''n''</sup>. A '''valuation''' is a function ''v'':'''K'''<sup>''n''</sup> → '''R''' such that ''v''(∅) = 0 and, for every ''S'',''T'' ∈'''K'''<sup>''n''</sup> for which ''S''∪''T''∈'''K'''<sup>''n''</sup>,
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| :<math> v(S) + v(T) = v(S \cap T) + v(S \cup T)~.</math>
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| A valuation is called continuous if it is continuous with respect to the [[Hausdorff metric]]. A valuation is called invariant under rigid motions if ''v''(''φ''(''S'')) = ''v''(''S'') whenever ''S'' ∈ '''K'''<sup>''n''</sup> and ''φ'' is either a [[translation (geometry)|translation]] or a [[rotation (mathematics)|rotation]] of '''R'''<sup>''n''</sup>.
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| ===Quermassintegrals===
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| {{main|quermassintegral}}
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| The quermassintegrals ''W''<sub>''j''</sub>: '''K'''<sup>''n''</sup> → '''R''' are defined via Steiner's formula
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| :<math> \mathrm{Vol}_n(K + t B) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j~,</math>
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| where ''B'' is the Euclidean ball. For example, ''W''<sub>0</sub> is the volume, ''W''<sub>''1''</sub> is proportional to the [[Minkowski content|surface measure]], ''W''<sub>''n''-1</sub> is proportional to the [[mean width]], and ''W''<sub>''n''</sub> is the constant Vol<sub>''n''</sub>(''B'').
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| ''W''<sub>''j''</sub> is a valuation which is [[homogeneous function|homogeneous]] of degree ''n''-''j'', that is,
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| :<math>W_j(tK) = t^{n-j} W_j(K)~, \quad t \geq 0~. </math> | |
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| ==Statement==
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| Any continuous valuation ''v'' on '''K'''<sup>''n''</sup> that is invariant under rigid motions can be represented as
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| :<math>v(S) = \sum_{j=0}^n c_j W_j(S)~.</math>
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| ===Corollary===
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| Any continuous valuation ''v'' on '''K'''<sup>''n''</sup> that is invariant under rigid motions and homogeneous of degree ''j'' is a multiple of ''W''<sub>''n''-''j''</sub>.
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| ==References==
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| An account and a proof of Hadwiger's theorem may be found in
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| * {{cite book|mr=1608265|last=Klain|first=D.A.|last2=Rota|author2-link=Gian-Carlo Rota|first2=G.-C.|title=Introduction to geometric probability|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-59362-X}}
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| An elementary and self-contained proof was given by Beifang Chen in
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| * {{cite journal|title=A simplified elementary proof of Hadwiger's volume theorem|journal=Geom. Dedicata|volume=105|year=2004|pages=107–120|last=Chen|first=B.|mr=2057247}}
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| [[Category:Integral geometry]]
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| [[Category:Theorems in convex geometry]]
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| [[Category:Probability theorems]]
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I'm a 38 years old, married and working at the university (Design and Technology).
In my free time I try to teach myself Swedish. I've been there and look forward to go there sometime near future. I like to read, preferably on my kindle. I like to watch How I Met Your Mother and Grey's Anatomy as well as docus about nature. I love Shooting sport.
Here is my website 4inkjets 15 Off Coupon Code