Supporting hyperplanes at relative boundary points

Fix affine coordinates $H\cong {\mathbb{R}}^k$. Closure preserves convexity without choosing approximating sequences: for $x,y\in K$ and any $ϵ>0$, choose two points of $C$ within $ϵ$ of them; their convex combination approximates $(1-t)x+ty$ within $ϵ$, so this point lies in $K$. Enumerate rational $k$-tuples canonically by increasing sums of absolute numerators and positive denominators, breaking the finite ties lexicographically. Density [F6] and the open complement of $K$ show that for each $m\in \mathbb{N}$ at least one enumerated point lies in $B(z,{ϵ}_m)\setminus K$. Let $z_m$ be the first such point. This is a uniquely specified sequence with $z_m\to z$, not a countable selection of arbitrary witnesses.

The nonempty compact set $K\cap \bar{B}(z_m,\mathrm{\parallel }{z}_m-z\mathrm{\parallel }+1)$ contains $z$. By [F2] and [F4], squared distance to $z_m$ attains a minimum there, and every point outside that ball is farther than the candidate $z$. Thus a nearest point $q_m\in K$ exists. It is unique: if two distinct minimizers had squared distance $d_m^2$, their midpoint would have squared distance $d_m^2-\mathrm{\parallel }{q}_m-q_m^{\mathrm{\prime }}{\mathrm{\parallel }}^2\mathrm{/}4$, contradicting minimality and convexity. Consequently $(q_m)$ is also canonically determined. Since $z_m\mathrm{\notin }K$, put $u_m=(z_m-q_m)\mathrm{/}\mathrm{\parallel }{z}_m-q_m\mathrm{\parallel }$. Expanding the distance at $q_m+t(q-q_m)$ and letting $t\downarrow 0$ gives $u_m\cdot (q-q_m)\le 0$ for every $q\in K$. Also $\mathrm{\parallel }{q}_m-z\mathrm{\parallel }\le 2\mathrm{\parallel }{z}_m-z\mathrm{\parallel }\to 0$.

Every $u_m$ has norm one. The exact choice-free sequential-compactness supplier [F5] gives a subsequence converging to a unit vector $u$. This introduces no additional choice hypothesis.

For every $q\in K$, pass to the limit in $u_m\cdot (q-q_m)\le 0$ along that subsequence. Continuity of the Euclidean pairing and $q_m\to z$ give $u\cdot (q-z)\le 0$. The unit vector $u$ defines a nonzero supporting functional for $K$, hence for $C$, at $z$.

If $C\subset {\mathbb{R}}^d$ has affine hull $H$ of dimension at most $d-2$, write its direction space as the span of $k\le d-2$ independent vectors $b_1,\dots ,b_k$. Its orthogonal complement is the solution space of the $k$ homogeneous equations $b_j\cdot n=0$. Row reduction has at most $k$ pivot variables, leaving at least $d-k\ge 2$ free variables. Assigning one free variable the value one and the others zero gives two independent solutions (their free-coordinate vectors are different unit vectors). Thus choose two nonparallel nonzero normals $n_1,n_2$ orthogonal to the direction space of $H$. For each $j$, the hyperplane $\{q:n_j\cdot (q-z)=0\}$ contains $H$ and therefore supports $C$ at $z$. The two hyperplanes are distinct because their normals are not parallel.