Partitioning Point Sets in 4 Dimensions
Partitioning Point Sets in 4 Dimensions
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In our eReader you can find the full English version of the book. Read Partitioning Point Sets in 4 Dimensions Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book Partitioning Point Sets in 4 Dimensions
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(in our construction 8 = 1/32, so that m = \/lOO suffices. ) Call this the continuous set.
Lemma 5: If there is a 8(n+m)-partition Q' of the continuous set then there b a fin-partition of the point set S.
Proof: Qearly a 8(n+m)-paitition of the continuous set implies a [8n-(l— 8)in]-partition of the sphere set, and hence of the point set 5. On a point set diis is a (f8n - (l-8)ml)-partition, which by assumption is a 8n-partition. □ Thus to obtain our result it suffices to show that a body S, of... mass n, has an /t/32-partition. We use the following two lonmas in our constnictioa.
Lemom 6: [W] Given a connected body S of mass n, in two dimensional Eudidean space, and given a line L forming an n/2-partition of S, there is a unique line K such that L and K form an n/4 partition. Furthermore, K varies continuously with L.
^^ard did not prove exactly this theoron, but the proof would be identical to Willard's. The fact that 5 is connected guarantees uniqueness.
Lemma 7: (Borsuk-Ulum theorem [L]).
What to read after Partitioning Point Sets in 4 Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by Richard Cole to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
(in our construction 8 = 1/32, so that m = \/lOO suffices. ) Call this the continuous set.
Lemma 5: If there is a 8(n+m)-partition Q' of the continuous set then there b a fin-partition of the point set S.
Proof: Qearly a 8(n+m)-paitition of the continuous set implies a [8n-(l— 8)in]-partition of the sphere set, and hence of the point set 5. On a point set diis is a (f8n - (l-8)ml)-partition, which by assumption is a 8n-partition. □ Thus to obtain our result it suffices to show that a body S, of... mass n, has an /t/32-partition. We use the following two lonmas in our constnictioa.
Lemom 6: [W] Given a connected body S of mass n, in two dimensional Eudidean space, and given a line L forming an n/2-partition of S, there is a unique line K such that L and K form an n/4 partition. Furthermore, K varies continuously with L.
^^ard did not prove exactly this theoron, but the proof would be identical to Willard's. The fact that 5 is connected guarantees uniqueness.
Lemma 7: (Borsuk-Ulum theorem [L]).
What to read after Partitioning Point Sets in 4 Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by Richard Cole to read onlineMoreLess
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