Abstract Let 𝔪 ∈ ℕ ${\mathfrak {m}\in \mathbb {N}}$ , 𝔪 ≥ 2 ${\mathfrak {m}\ge 2}$ , and let { p j } j = 1 𝔪 ${\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ be a finite subset of 𝕊2 such that 0 → ∈ ℝ 3 ${\vec{0}\in \mathbb {R}^3}$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies 𝒦 in ℝ3 whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and 𝔪 planar compact discs D ¯ 1 , ... , D ¯ 𝔪 ${\overline{D}_1,\ldots ,\overline{D}_\mathfrak {m}}$ , such that the Gauss map of S is a homeomorphism onto 𝕊 2 - { p j } j = 1 𝔪 ${\mathbb {S}^2-\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ and D j ⊥ p j ${D_j\bot p_j}$ , for all j . ${j.}$ We derive applications to existence of harmonic diffeomorphisms between domains of 𝕊2, existence of capillary surfaces in ℝ3, and a Hessian equation of Monge–Ampère type.