| The capacity of bounded, convex regions in ${bf R}sp{n}$ is computed up to dimensional constants. The first variation of such capacities is then investigated. For regions in ${bf R}sp{n}$ with $nge 3,$ the variation of capacity in $n{-}2$ mutually orthogonal directions is computed up to dimensional constants. In the remaining two directions, certain bounds on the variation of capacity are obtained. It is shown that the lower bounds found are sharp (up to dimensional constants) by computing explicitly the variation of capacity for ellipsoids.;In the case of $n = 2,$ the variation of capacity is computed up to constants in both directions. In addition, for regions in ${bf R}sp{n}$ which have a uniform upper bound on the eccentricities of their cross sections perpendicular to a diameter, the variation of capacity is computed up to dimensional constants in all directions. Finally, a conjecture is made which describes the variation of capacity for any convex region, up to dimensional constants. |
