  It is known that any subharmonic quadrature domain in two dimensions
satisfies a natural inner ball condition, in other words there is a specific
upper bound on the curvature of the boundary. This result directly applies to
free boundaries appearing in obstacle type problems and in Hele-Shaw flow. In
the present paper we make partial progress on the corresponding question in
higher dimensions. Specifically, we prove the equivalence between several
different ways to formulate the inner ball condition, and we compute the
Brouwer degree for a geometrically important mapping related to the Schwarz
potential of the boundary. The latter gives in particular a new proof in the
two dimensional case.
