On some isoperimetric inequalities for the Newtonian capacity

Upper bounds are obtained for the Newtonian capacity of compact sets in Rd,d≥3ℝd,d≥3 in terms of the perimeter of the rr-parallel neighborhood of KK. For compact, convex sets in Rd,d≥3ℝd,d≥3 with a C2C2 boundary the Newtonian capacity is bounded from above by (d−2)M(K)(d−2)M(K), where M(K)>0M(K)>0 is the integral of the mean curvature over the boundary of KK with equality if KK is a ball. For compact, convex sets in Rd,d≥3ℝd,d≥3 with non-empty interior the Newtonian capacity is bounded from above by (d−2)P(K)2d|K|(d−2)P(K)2d|K| with equality if KK is a ball. Here, P(K)P(K) is the perimeter of KK and |K||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 Rd,d≥5ℝd,d≥5 with radius ε𝜀 and time length tt.