In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\mbox{Cap}_{\mathcal{A}}$, where $\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the $p$-Laplace equation and whose solutions in an open set are called $ \mathcal{A}$-harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ \left[\mbox{Cap}_\mathcal{A}(\lambda E_1 +(1-\lambda)E_2)\right]^{\frac{1}{(n-p)}}\geq\lambda\left[\mbox{Cap}_\mathcal{A}(E_1)\right]^{\frac{1}{(n-p)}}+(1-\lambda)\left[\mbox{Cap}_\mathcal{A}(E_2 )\right]^{\frac{1}{(n-p)}} \] when $1Comment: 108 pages. Some small typos were corrected, some minor changes and a new lemma (Lemma 10.2) in section 10
