Lower bounds for the volume of the image of a ball

Abstract

We consider ring $Q$-homeomorphisms with respect to $p$-modulus in the space $\Bbb R^{n}$ as $p>n$. We obtain a lower bound for the
volume of the image of a ball under such mappings. We solve the extremal problems on minimizing of the functionals of the volume of
the image of a ball and the area of the image of a sphere.