Download American Mathematical Monthly, volume 117, number 2, by Daniel J. Velleman PDF

By Daniel J. Velleman

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Extra resources for American Mathematical Monthly, volume 117, number 2, February 2010

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Comparison of K (d, ∞) obtained via the three proof methods: Medium dashing (bottom) = Nemirovski; Small and tiny dashing (middle) = type 2 inequalities; Large dashing (top) = truncation and Bernstein inequality. February 2010] NEMIROVSKI’S INEQUALITIES REVISITED 149 6. PROOFS. 1. Proofs for Section 2. Proof of (6). In the case r = ∞, the asserted inequalities read x ∞ ≤ x q ≤ d 1/q x ∞ for 1 ≤ q < ∞ and are rather obvious. For 1 ≤ q < r < ∞, (6) is an easy consequence of H¨older’s inequality. 4.

The three required properties needed to show that the curve is self-avoiding are satisified exactly when π − θ > 2φ. After some computation, this inequality is shown to be equivalent to c< 1 2 tan2 (θ/2) +1 . If n is odd, then approximate the curve AC by an isoceles trapezoid with base AC. In order to define the top side of the trapezoid, consider the line that is parallel to the top left segment of the primary n-gon and goes through the top left vertex of the primary n-gon thereof (see Figure 8).

However, the self-similarity of the curve greatly reduces the amount of work that needs to be done. First of all, it is superfluous to search for intersections between every pair of points on the curve. 1). This region is illustrated in Figures 5 and 9. S M C Figure 9. 037, with circular approximations. The grayed circles do not need to be checked by self-similarity. Furthermore, within the wedge shaped region between segments M S and MC, there is a considerable amount of repetition. In order to simplify the search for selfintersection, this repetition needs to be eliminated, and so we introduce the rescaling map that fixes the point M and shrinks all distances by a factor of (1 − c)/2.

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