Lecture7Notes

Lecture7Notes - C&O 355 Fall 2010 Lecture 7 Notes...

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Unformatted text preview: C&O 355, Fall 2010 Lecture 7 Notes Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Covering Hemispheres by Ellipsoids Recall our notation B = { x : k x k ≤ 1 } and H u = x : x T u ≥ , where u is an arbitrary unit vector. The next theorem defines an ellipsoid that covers B ∩ H u and analyzes its volume. For simplicity, let us assume that n ≥ 3. Theorem 1. Define M = n 2 n 2- 1 I- 2 n + 1 uu T b = u n + 1 B = E ( M,b ) = n x : ( x- b ) T M- 1 ( x- b ) ≤ 1 o . Then B satisfies the following two properties. B ∩ H u ⊆ B (1) vol( B ) vol( B ) ≤ e- 1 / 4( n +1) ≤ 1- 1 8( n + 1) (2) The following two claims prove the theorem. Claim 2. B ∩ H u ⊆ B . Proof. First note that we can derive an explicit expression for M- 1 using our Claim 1 on rank-1 updates. M- 1 = n 2- 1 n 2 I + 2 n- 1 uu T Substitute this into the definition of E ( M,b ): B = x : x- u n + 1 T n 2- 1 n 2 I + 2 n- 1 uu T x- u n + 1 ≤ 1 = x : x- u n + 1 T x- u n + 1 + 2 n- 1 u T x- u n + 1 !...
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.

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Lecture7Notes - C&O 355 Fall 2010 Lecture 7 Notes...

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