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In this article we study two classical problems in convex geometry associated to{\mathcal{A}}   -harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the  p -Laplace equation. Let  p  be fixed with2  ≤  n  ≤  p  &lt;  ∞{2\leq n\leq p&lt;\infty}   . For a convex compact set  E  inℝ  n{\mathbb{R}^{n}}   , we define and then prove the existence and uniqueness of the so-called{\mathcal{A}}   -harmonic Green’s function for the complement of  E  with pole at infinity. We then define a quantityC     ⁢   (  E  ){\mathrm{C}_{\mathcal{A}}(E)}    which can be seen as the behavior of this function near infinity. In the first part of this article, we prove thatC     ⁢   (  ⋅  ){\mathrm{C}_{\mathcal{A}}(\,\cdot\,)}    satisfies the following Brunn–Minkowski-type inequality:

The Brunn--Minkowski inequality and a Minkowski problem for 𝒜-harmonic Green's function