  This paper is concerned with stability of the ball for a class of
isoperimetric problems under convexity constraint. Considering the problem of
minimizing $P+\varepsilon R$ among convex subsets of $\mathbb{R}^N$ of fixed
volume, where $P$ is the perimeter functional, $R$ is a perturbative term and
$\varepsilon>0$ is a small parameter, stability of the ball for this perturbed
isoperimetric problem means that the ball is the unique (local, up to
translation) minimizer for any $\varepsilon$ sufficiently small. We investigate
independently two specific cases where $\Omega\mapsto R(\Omega)$ is an energy
arising from PDE theory, namely the capacity and the first Dirichlet eigenvalue
of a domain $\Omega\subset\mathbb{R}^N$. While in both cases stability fails
among all shapes, in the first case we prove (non-sharp) stability of the ball
among convex shapes, by building an appropriate competitor for the capacity of
a perturbation of the ball. In the second case we prove sharp stability of the
ball among convex shapes by providing the optimal range of $\varepsilon$ such
that stability holds, relying on the \emph{selection principle} technique and a
regularity theory under convexity constraint.
