Abstract:

Gromov's celebrated systolic inequality from '83 is a universal volume lower bound for an essential manifold M in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every C>0. Our key tool is the first variation formula thought of as a real statement in first-order logic, in the context of non-standard analysis.