Gromov's celebrated systolic inequality from '83 is a universal volume lower bound 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+ε)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ε > 0, using the first variation formula and the mean value theorem.
Bibliographical noteFunding Information:
Supported by the Israel Science Foundation (grants no. 84/03 and 1294/06) and the BSF (grant 2006393).
- Essential manifold
- Finite-dimensional approximation
- First variation formula
- Gromov's inequality
- Injectivity radius
- Kuratowski imbedding
- Systolic inequality