On some isoperimetric inequalities for the Newtonian capacity

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Abstract
Upper bounds are obtained for the Newtonian capacity of compact sets in [ ] in terms of the perimeter of the [ ]-parallel neighborhood of [ ]. For compact, convex sets in [ ] with a [ ] boundary the Newtonian capacity is bounded from above by [ ], where [ ] is the integral of the mean curvature over the boundary of [ ] with equality if [ ] is a ball. For compact, convex sets in [ ] with non-empty interior the Newtonian capacity is bounded from above by [ ] with equality if [ ] is a ball. Here, [ ] is the perimeter of [ ] and [ ] is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in [ ] with radius [ ] and time length [ ].

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