On some isoperimetric inequalities for the Newtonian capacity

Upper bounds are obtained for the Newtonian capacity of compact sets in $\R^d,\,d\ge 3$ in terms of the perimeter of the r-parallel neighbourhood of K. For compact, convex sets in $\R^d,\,d\ge 3$ with a C2 boundary the Newtonian capacity is bounded from above by (d−2)M(K), where M(K)>0 is the integral of the mean curvature over the boundary of K with equality if K is a ball. For compact, convex sets in $\R^d,\,d\ge 3$ with non-empty interior the Newtonian capacity is bounded from above by (d−2)P(K)2d|K| with equality if K is a ball. Here P(K) is the perimeter of K and |K| 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 $\R^d,\,d\ge 5$ with radius ε and time length t.