Measurable minimum-norm geometric-median selection

Let $V=\mathrm{span}\{m_1,\dots ,m_k\}$. In the given center order, Gram–Schmidt replaces $m_j$ by $r_j=m_j-{\sum }_{\mathrm{\ell }<j}⟨m_j,e_{\mathrm{\ell }}⟩e_{\mathrm{\ell }}$ using the previously retained orthonormal vectors; if $r_j=0$ it is skipped, and otherwise $e=r_j\mathrm{/}\mathrm{\parallel }{r}_j\mathrm{\parallel }$ is appended. Induction on the finite list shows the retained vectors are orthonormal and span $V$. For $d=\dim V>0$, if $v_n={\sum }_{\mathrm{\ell }=1}^d{a}_{n\mathrm{\ell }}{e}_{\mathrm{\ell }}\in V$ converges in $H$ to $v$, then [F4] gives $a_{n\mathrm{\ell }}=⟨v_n,e_{\mathrm{\ell }}⟩\to ⟨v,e_{\mathrm{\ell }}⟩$ for each $\mathrm{\ell }$; the finite sums converge to ${\sum }_{\mathrm{\ell }}⟨v,e_{\mathrm{\ell }}⟩e_{\mathrm{\ell }}=v$, so $V$ is closed. If $d=0$, all centers are zero and $F_m(z)=k\mathrm{\parallel }z\mathrm{\parallel }$ has the unique minimizer $0$. For $d>0$, set $P_{V}z={\sum }_{\mathrm{\ell }}⟨z,e_{\mathrm{\ell }}⟩e_{\mathrm{\ell }}$; its residual is orthogonal to $V$. Pythagoras gives $\mathrm{\parallel }z-m_j{\mathrm{\parallel }}^2=\mathrm{\parallel }{P}_{V}z-m_j{\mathrm{\parallel }}^2+\mathrm{\parallel }z-P_{V}z{\mathrm{\parallel }}^2$, so $F_m(P_{V}z)<F_m(z)$ whenever $z\mathrm{\notin }V$. Thus every minimizer lies in $V$. The function $F_m$ is continuous and coercive because $F_m(z)\ge k\mathrm{\parallel }z\mathrm{\parallel }-{\sum }_j\mathrm{\parallel }{m}_j\mathrm{\parallel }$; its sublevel set $\{z\in V:F_m(z)\le F_m(0)\}$ is nonempty, closed and bounded in the finite-dimensional coordinate space ${\mathbb{R}}^d$, hence compact by [F3]. By [F5], $F_m$ attains a minimum there; the full minimizer set $C_m$ is nonempty, closed, bounded and compact, and it is convex because $F_m$ is convex.

By [F2], $C_m$ has a unique point $z_{\ast }(m)$ nearest to $0$, its unique minimum-norm point. If $H=\{0\}$, the selection is the constant zero map; otherwise fix a unit vector $e$ and a countable dense sequence $(d_a{)}_{a\ge 1}$ in $H$, whose existence is given by [F8]. The countable set $D=\{⟨d_a,e⟩:a\ge 1\}$ is dense in $\mathbb{R}$: for each $t\in \mathbb{R}$, density gives $d_{a_j}\to te$, and [F4] gives $⟨d_{a_j},e⟩\to t$. For $\lambda >0$, $G_{\lambda ,m}(z)=F_m(z)+\lambda \mathrm{\parallel }z{\mathrm{\parallel }}^2$ is continuous, coercive and strictly convex. Projection to $V$ cannot increase $F_m$ and strictly decreases the squared norm off $V$, so every minimizer lies in $V$; if $d=0$ the single candidate is $0$. For $d>0$, a closed bounded sublevel set in $V$ is compact by [F3], so [F5] gives a unique minimizer $z_{\lambda }(m)$. By continuity and density, $h_{\lambda }(m)={\inf }_a{G}_{\lambda ,m}(d_a)=G_{\lambda ,m}(z_{\lambda }(m))$. Each map $m\mapsto G_{\lambda ,m}(d_a)$ is continuous, so [F7] makes their countable infimum $h_{\lambda }$ Borel. For $s\ge 1$, set $A_{a,s}=\{m:G_{\lambda ,m}(d_a)<h_{\lambda }(m)+1\mathrm{/}s\}$. The countable dense set $D$ gives $A_{a,s}={\bigcup }_{q\in D}\{m:G_{\lambda ,m}(d_a)<q\}\cap \{m:h_{\lambda }(m)>q-1\mathrm{/}s\}$ so $A_{a,s}$ is Borel; density ensures at least one such $a$ for each $m$. The least such $a$ is a Borel, countably-valued map because its level set at $a$ is $A_{a,s}\setminus {\bigcup }_{b<a}{A}_{b,s}$, and the selected point map is Borel since each open-set preimage is the union of its Borel level sets whose values lie in that open set. The selected point is within $(\lambda s{)}^{-1\mathrm{/}2}$ of $z_{\lambda }(m)$: convexity of $F_m$ and $\mathrm{\parallel }(1-t)z+ty{\mathrm{\parallel }}^2=(1-t)\mathrm{\parallel }z{\mathrm{\parallel }}^2+t\mathrm{\parallel }y{\mathrm{\parallel }}^2-t(1-t)\mathrm{\parallel }y-z{\mathrm{\parallel }}^2$, applied at the minimizer and followed by $t\downarrow 0$, give $G_{\lambda ,m}(y)-G_{\lambda ,m}(z_{\lambda })\ge \lambda \mathrm{\parallel }y-z_{\lambda }{\mathrm{\parallel }}^2$. Hence the selected maps converge pointwise to $z_{\lambda }$. Since they are countably valued with Borel level sets, their distances to each $d_a$ are Borel and converge to $\mathrm{\parallel }{z}_{\lambda }-d_a\mathrm{\parallel }$; [F7] makes these limits Borel. Balls with centers $d_a$ and positive rational radii form a countable basis of the separable metric space $H$: every open neighborhood of a point contains such a ball around a sufficiently close dense center. Thus inverse images of open sets are countable unions of Borel inverse images of these balls, proving $m\mapsto z_{\lambda }(m)$ is Borel.

For every $s\ge 1$, comparison with $z_{\ast }(m)$ in the minimization of $F_m+s^{-1}\mathrm{\parallel }\cdot {\mathrm{\parallel }}^2$ gives $\mathrm{\parallel }{z}_{1\mathrm{/}s}\mathrm{\parallel }\le \mathrm{\parallel }{z}_{\ast }\mathrm{\parallel }$ and $F_m(z_{1\mathrm{/}s})\le F_m(z_{\ast })+s^{-1}\mathrm{\parallel }{z}_{\ast }{\mathrm{\parallel }}^2$. If $d=0$ then both points are $0$. Otherwise all $z_{1\mathrm{/}s}$ lie in the compact ball $B_V(0,\mathrm{\parallel }{z}_{\ast }\mathrm{\parallel })$ by [F3]. The closed tail sets $C_n=\stackrel{‾}{\{z_{1\mathrm{/}s}:s\ge n\}}$ inside this ball are nonempty, nested and have the finite-intersection property, so [F6] gives $c\in {\bigcap }_n{C}_n$. For every $n$, continuity of $F_m$ and the tail bounds imply $F_m(c)\le F_m(z_{\ast })+\mathrm{\parallel }{z}_{\ast }{\mathrm{\parallel }}^2\mathrm{/}n$, while $\mathrm{\parallel }c\mathrm{\parallel }\le \mathrm{\parallel }{z}_{\ast }\mathrm{\parallel }$; letting $n\to \mathrm{\infty }$ shows $c\in C_m$ is a minimum-norm minimizer, hence $c=z_{\ast }$. If the sequence did not converge to $z_{\ast }$, some $\rho >0$ would leave arbitrarily late terms at distance at least $\rho$; the closed tail closures of those terms are nonempty nested subsets of the same compact ball, so [F6] gives a point in their intersection. It lies in every $C_n$, hence equals $z_{\ast }$, but its distance from $z_{\ast }$ is at least $\rho$, a contradiction. Thus $z_{1\mathrm{/}s}\to z_{\ast }$. By [F7] and the separable-metric basis argument of step 1.2, the pointwise limit of the Borel maps $z_{1\mathrm{/}s}$ is Borel.

The minimizer set is nonempty, closed and convex by step 1.1, its minimum-norm point is unique by step 1.2, and step 2.1 proves the selected map is Borel. The only choice principle used is Countable Choice in [F2]; the dense sequence is supplied by separability and all subsequent approximations use least-index rules.