The rectangular Mazur-distance objective attains its minimum on the Schatten power sphere

Let $E,F$ be finite-dimensional complex inner-product spaces, let $\kappa$ be a finite index type and $e$ an orthonormal basis of $E$ indexed by $\kappa$, let $p\in\mathbb R$ with $p>0$, and let $\iota$ be a nonempty finite index type. For a complex-linear map $T:E\to F$ with singular values $\operatorname{sv}_k(T)$, the Schatten $p$-power sphere is $\{T\mid\sum_k\operatorname{sv}_k(T)^p=1\}$ (schattenPowerSphere). For $T:E\to F$ write $\widehat T$ for its Hermitian dilation, the self-adjoint operator on $E\oplus F$ with $\widehat T(x,y)=(T^\ast y,\,Tx)$ (hermitianDilation), and let $\psi_p(x)=\operatorname{sign}(x)\,|x|^{p/2}$ be the scalar Mazur map (scalarMazur). Applying $\psi_p$ to the eigenvalues of $\widehat T$ in an orthonormal eigenbasis gives a self-adjoint operator that is again the dilation of a unique complex-linear map $E\to F$; that map is the rectangular Mazur map $\Psi_p(T)$ (rectangularMazurMap), so $\widehat{\Psi_p(T)}=\psi_p(\widehat T)$. It satisfies $\sum_k\operatorname{sv}_k(\Psi_p(T))^2=\sum_k\operatorname{sv}_k(T)^p$, so it sends the Schatten $p$-power sphere into the ordinary Schatten-$2$ (Hilbert–Schmidt) unit sphere.

Fix real weights $a:\iota\to\mathbb R$ (no sign condition on the $a_i$) and a family $(u_i)_{i\in\iota}$ on the Schatten $p$-power sphere. For $v$ ranging over the same sphere, consider the weighted objective

with $\|\cdot\|_2$ the Schatten-$2$ norm, and let $w=\sum_{i\in\iota}a_i\,\Psi_p(u_i)$ be the weighted barycenter of the Mazur images (rectangularMazurBarycenter). Then the theorem states that $f$ attains the global minimum value

over the Schatten $p$-power sphere: this value lower-bounds $f(v)$ for every $v$ on the sphere, and equals $f(v)$ for some $v$ on the sphere.

This is the Hilbert-side half of the variational comparison: it computes, in closed form, the minimum of the weighted squared-distance objective on the actual Schatten power sphere of rectangular operators — with no enlargement to an ambient coordinate space — matching the identity $\min_{\|v\|=1}\sum_ia_i\|u_i-v\|^2=2\big(\sum_ia_i-\|\sum_ia_iu_i\|\big)$ for unit vectors $u_i$ in an ordinary Hilbert space.

Formalization Note The statement is additionally parametrized by an orthonormal basis $e$ of $E$ indexed by $\kappa$, used only to identify Hilbert–Schmidt coordinates on $E$ in the proof; neither the objective, the barycenter, nor the minimum value depends on the choice of $e$.