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In this paper we give several results based on randomized embeddings of l2 into l∞(or "l∞-like") spaces. Our first result is a (1 + ε)-distortion asymmetric embedding of n points in l2 into l∞ with polylog(n) dimension, for any 1 + ε. This gives the first known O(1)- approximate nearest neighbor algorithm with fast query time and almost polynomial space for a product of Euclidean norms, a common generalization of both l2 and l∞ norms. Our embedding also clarifies the relative complexity of approximate nearest neighbor in l2 and l∞ spaces.Our second result in a (1 + ε)-approximate algorithm for the diameter of n points in ld2, running in time Õ(dn1+l/(1+ε)2); the algorithm is fully dynamic. This improves several previous algorithms for this problem (see Table 1 for more information).