For p ε (2;∞), the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this re-sult include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ε (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m, n ε N, the small-est possible bi-Lipschitz distortion of any embedding into Lp of the grid (1,...m)n⊆ lnq is bounded above and below by constant multiples (de-pending only on p; q) of the quantity min(n(p-q)(q-2)/(q2(p-2));m(q-2)/q).
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty