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We show that any well-behaved demand function can be represented by its "demand manifold," a smooth curve that relates the elasticity and convexity of demand. This manifold is a sufficient statistic for many comparative statics questions; leads naturally to characterizations of new families of demand functions that nest most of those used in applied economics; and connects assumptions about demand structure with firm behavior and economic performance. In particular, the demand manifold leads to new insights about industry adjustment with heterogeneous firms, and can be empirically estimated to provide a quantitative framework for measuring the effects of globalization.