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

By Daniel J. Velleman

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First, the polygon encloses the base and primary n-gon of the curve. Second, the approximation of each of the curve’s children fits entirely within the curve’s approximation. Third, all of the approximations of the curve’s children are pairwise disjoint (except for adjacent children, whose approximations are disjoint save for the shared vertex). By self-similarity, these three properties ensure that the curve is selfavoiding. February 2010] VON KOCH CURVES BUILT FROM n-GONS 129 π–θ S φ M A C Figure 7.

Rademacher Type and Cotype Inequalities. Let { i } denote a sequence of independent Rademacher random variables. Let 1 ≤ p < ∞. A Banach space B with norm · is said to be of (Rademacher) type p if there is a constant T p such that for all finite sequences {xi } in B, p n E n ≤ T pp i xi p xi i=1 . i=1 Similarly, for 1 ≤ q < ∞, B is of (Rademacher) cotype q if there is a constant Cq such that for all finite sequences {xi } in B, q n E i xi n ≥ Cq−q xi i=1 q . i=1 Ledoux and Talagrand [13, p. 247] note that type and cotype properties appear as dual notions: if a Banach space B is of type p, its dual space B is of cotype q = p/( p − 1).

2 entails that for large dimension d, the constants K Nem (d, ∞) and . 7183. 2. Arbitrary Lr -spaces. 1 is a special case of a more general inequality: Let (T, , μ) be a σ -finite measure space, and for 1 ≤ r < ∞ let L r (μ) be the set of all measurable functions f : T → R with finite norm f r := | f |r dμ 1/r , where two such functions are viewed as equivalent if they coincide almost everywhere with respect to μ. In what follows we investigate the functional f → V ( f ) := f 2 r on L r (μ). Note that (Rd , · r ) corresponds to (L r (μ), · r ) if we take T = {1, 2, .

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American Mathematical Monthly, volume 117, number 2, February 2010 by Daniel J. Velleman

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