 Solution

For a finite nonempty collection $X=\{p_1,\dots ,p_n\}\subset {\mathbb{R}}^3$, its convex hull is the smallest convex set containing all the points. Equivalently, it is the set of all convex combinations:Every convex set containing the $p_i$ must contain these combinations; the displayed set is itself convex and contains each $p_i$, proving the equivalence. For a full-dimensional finite collection this is a convex polyhedron. A collinear or coplanar collection instead has a lower-dimensional convex hull, so the solid case needs a separate dimensionality assumption.