Macroscopic band width inequalities

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce new class orientable manifolds call filling enlargeable and prove: If $M$ all unit balls in universal cover $(V,g)$ have volume less than constant $\frac{1}{2}\varepsilon_n$, then $width(V,g)\leq1$. show that if manifold or aspherical, it enlargeable. Furthermore whether not only depends image fundamental under classifying map cover.