The mapping properties of filling radius and packing radius and their applications (Q1765692)

The filling radius \(\text{FillRad}(V)\) of a closed, orientable Riemannian manifold \(V\) is defined via the canonical isometric embedding \(J:V\to L^\infty(V)\) into the Banach space of bounded Borel functions given by \(J(v)=\text{dist}(v,\cdot)\). Then, \(\text{FillRad}(V)\) is the smallest \(\varepsilon\) such that the (integer) fundamental class of \(V\) vanishes in the \(\varepsilon\)-neighborhood of \(J(V)\) in \(L^\infty(V)\). This invariant was introduced by \textit{M.~Gromov} in the paper [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)]. The main result of the paper is the inequality \[ \text{FillRad}(W)\leq\text{Lip}(f)\cdot \text{FillRad}(V) \] whenever there is a Lipschitz map \(f:V\to W\) of degree \(\pm 1\); here \(\text{Lip}(f)\) is the Lipschitz constant of \(f\). The proof is based on the existence of a Lipschitz extension \(\widetilde f:L^\infty(V)\to L^\infty(W)\) of the Lipschitz map \(f:J(V)\to J(W)\) with the same Lipschitz constant which is explained in detail. A number of applications for estimates of the filling radius from below and for a mapping property of the packing radii is presented.