  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 $C^2$ 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 $\frac{(d-2)P(K)^2}{d|K|}$ 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
$\varepsilon$ and time length $t$.
