  The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and
$B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$.
Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets
in $\mathbb{R}^d$. The concept of stability in this context concerns how, when
approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In
a recent breakthrough [FvHT23], the authors of this paper proved the following
folklore conjectures on the sharp stability for the Brunn-Minkowski inequality:
  (1) A linear stability result concerning the distance from $A$ and $B$ to
their respective convex hulls.
  (2) A quadratic stability result concerning the distance from $A$ and $B$ to
their common convex hull.
  As announced in [FvHT23], in the present paper, we leverage (1) in
conjunction with a novel optimal transportation approach to offer an
alternative proof for (2).
