  The aim of this paper is twofold. In the first part of this paper we focus on
a functional involving a weighted curvature integral and the quermassintegrals.
We prove upper and lower bounds for this functional in the class of convex
sets, which provide a stronger form of the classical Aleksandrov-Fenchel
inequality involving the $(n-1)$ and $(n-2)$-quermassintegrals, and
consequently a stronger form of the classical isoperimetric inequality in the
planar case. Moreover quantitative estimates are proved. In the second part we
deal with a shape optimization problem of a functional involving the boundary
momentum. It is known that in dimension two the ball is a maximizer among
simply connected sets when the perimeter and centroid is fixed. In higher
dimensions the same result does not hold and we consider a new scaling
invariant functional that might be a good candidate to generalize the
bidimensional case. For this functional we prove that the ball is a stable
maximizer in the class of nearly spherical sets in any dimension.
