---
title: Strongly Anti-Coproximinal Subspaces
url: https://www.emergentmind.com/topics/strongly-anti-coproximinal-subspaces
type: topic
---

# Strongly Anti-Coproximinal Subspaces

Searching arXiv for recent and foundational papers on strongly anti-coproximinal subspaces, anti-coproximinality, and related coproximinality/proximinality.
Strongly anti-coproximinal subspaces are subspaces \(Y\) of a Banach space \(X\) for which the best coapproximation problem fails in the strongest approximate sense: for every \(x\in X\setminus Y\) and every \(\varepsilon\in[0,1)\), there is no \(\varepsilon\)-best coapproximation to \(x\) out of \(Y\). The notion strengthens anti-coproximinality, is formulated through approximate Birkhoff–James orthogonality, and has developed into a geometric theory connecting support functionals, facial structure of the unit ball, and concrete classifications in finite-dimensional, function, and operator spaces [2407.14471, 2508.04971].

## 1. Definitions and basic framework

Let \(X\) be a real Banach space and \(Y\subset X\) a subspace. A vector \(y_0\in Y\) is a best coapproximation to \(x\in X\) out of \(Y\) if
\[
\|y_0-y\|\le \|x-y\| \qquad \text{for all } y\in Y.
\]
The set of all best coapproximations is denoted \(R_Y(x)\). In contrast with best approximation, \(R_Y(x)\) may be empty even in finite-dimensional settings [2407.14471].

For \(\varepsilon\in[0,1)\), \(y_0\in Y\) is an \(\varepsilon\)-best coapproximation to \(x\) out of \(Y\) if
\[
Y \perp_B^\varepsilon (x-y_0),
\]
that is, \(y\perp_B^\varepsilon (x-y_0)\) for every \(y\in Y\). Here approximate Birkhoff–James orthogonality is given by
\[
x\perp_B^\varepsilon y
\quad \Longleftrightarrow \quad
\|x+\lambda y\|\ge \|x\|-\varepsilon |\lambda|\,\|y\|
\quad \forall \lambda\in \mathbb R.
\]

A subspace \(Y\) is coproximinal if \(R_Y(x)\neq \varnothing\) for every \(x\in X\). It is anti-coproximinal if
\[
x\in X\setminus Y \implies R_Y(x)=\varnothing.
\]
It is strongly anti-coproximinal if for every \(x\in X\setminus Y\) and every \(\varepsilon\in[0,1)\), there is no \(\varepsilon\)-best coapproximation to \(x\) out of \(Y\). Thus
\[
\text{strongly anti-coproximinal} \implies \text{anti-coproximinal},
\]
but not conversely [2407.14471].

The property is relative to the ambient space. A subspace may be strongly anti-coproximinal in one superspace and fail even to be anti-coproximinal in a larger one. This ambient dependence is explicit in the finite-dimensional \(\ell_1\)-based examples discussed in the literature [2407.14471].

## 2. Orthogonality and duality formulations

The central structural identity is that best coapproximation is an orthogonality condition:
\[
y_0\in Y \text{ is a best coapproximation to } x
\quad \Longleftrightarrow \quad
Y\perp_B (x-y_0).
\]
Accordingly, strong anti-coproximinality can be read as the global failure of approximate orthogonality:
\[
Y \text{ strongly anti-coproximinal}
\quad \Longleftrightarrow \quad
Y\not\perp_B^\varepsilon x
\quad \text{for every nonzero } x\in X \text{ and every } \varepsilon\in[0,1).
\]
This places the subject squarely in the geometry of Birkhoff–James orthogonality [2407.14471].

If
\[
J(x):=\{f\in S_{X^*}: f(x)=\|x\|\},
\]
then the standard dual criterion for approximate orthogonality is
\[
x\perp_B^\varepsilon y
\quad \Longleftrightarrow \quad
\exists f\in J(x)\text{ such that } |f(y)|\le \varepsilon \|y\|.
\]
For \(\varepsilon\)-best coapproximation, this yields the equivalent condition: \(y_0\in Y\) is an \(\varepsilon\)-best coapproximation to \(x\) iff for every \(y\in Y\), there exists \(f_y\in J(y)\) such that
\[
|f_y(x-y_0)|\le \varepsilon \|x-y_0\|.
\]
At \(\varepsilon=0\), one recovers the exact coapproximation criterion [2407.14471].

A more abstract dual description uses selection maps. For a subspace \(\mathbb Y\subset \mathbb X\), a selection map is a map
\[
\phi:\mathbb Y\to \mathbb X^*,\qquad \phi(y)\in J(y),
\]
with the natural homogeneity condition \(f_{\mu y}=\overline{\mu}\,f_y\). Then anti-coproximinality is equivalent to
\[
\overline{\operatorname{span}(\operatorname{Im}\phi)}^{\,w^*}=\mathbb X^*
\qquad \forall \phi\in \Lambda_{\mathbb Y}.
\]
Under the dual-ball hypothesis
\[
B_{\mathbb X^*}=\overline{\operatorname{co}\big(w^*stExp(B_{\mathbb X^*})\big)}^{\,w^*},
\]
a closed proper subspace \(\mathbb Y\) is strongly anti-coproximinal iff for every selection map \(\phi\),
\[
w^*stExp(B_{\mathbb X^*})\subset \overline{\operatorname{Im}\phi}^{\,w^*}.
\]
This shows that anti-coproximinality is a weak-\(^*\) spanning condition, whereas strong anti-coproximinality requires approximation of the weak-\(^*\)-strongly exposed boundary of the dual ball by support functionals coming from the subspace [2508.04971].

## 3. General structural results in Banach spaces

A general sufficient condition for strong anti-coproximinality is available in arbitrary Banach spaces. If for each \(x\in X\setminus Y\) there exists \(y\in Y\) such that
\[
J(y)\subset J(x)\cup J(-x),
\]
then \(Y\) is strongly anti-coproximinal. The condition requires a strong alignment of support functionals of \(y\) with those of \(x\) or \(-x\), and it rules out all approximate coapproximants with \(\varepsilon<1\) [2407.14471].

A general necessary condition appears in the reflexive/Kadets–Klee setting. If \(X\) is reflexive, \(X^*\) has the Kadets–Klee property, \(Y\) is closed, and \(Y\) is strongly anti-coproximinal, then for each \(x\in X\) there exists \(y\in Y\) such that
\[
J(y)\cap J(x)\neq \varnothing.
\]
This necessary condition is not sufficient; explicit counterexamples show that shared support functionals do not by themselves force strong anti-coproximinality [2407.14471].

The theory is especially rigid for closed subspaces in spaces with strong convexity or smoothness properties. Every dense subspace of a Banach space is strongly anti-coproximinal, so the delicate part of the theory concerns closed subspaces. On the negative side, there are no closed strongly anti-coproximinal subspaces in broad smooth or strictly convex classes. In particular, no closed subspace is strongly anti-coproximinal if \(X\) is finite-dimensional smooth, finite-dimensional strictly convex, or uniformly smooth; more generally, every reflexive strictly convex Banach space whose dual has Kadets–Klee, and every reflexive smooth Banach space whose dual has Kadets–Klee, has no strongly anti-coproximinal closed subspaces [2407.14471].

The geometry of the ambient unit ball yields further necessary conditions. If \(Y\) is a closed proper strongly anti-coproximinal subspace, then \(Y\) must contain all w-ALUR points of \(X\). For finite-dimensional \(Y\), strong anti-coproximinality implies that \(Y\) intersects every maximal face of \(B_X\). A stronger sufficient condition is also known: if \(Y\) intersects the relative interior of every facet of \(B_X\), then \(Y\) is strongly anti-coproximinal. In finite-dimensional polyhedral Banach spaces these conditions become exact: \(Y\) is strongly anti-coproximinal iff \(Y\) intersects the interior of every facet of \(B_X\), equivalently iff
\[
J_Y=\operatorname{Ext}(B_{X^*}),
\]
where
\[
J_Y=\{f\in S_{X^*}: f(y)=1 \text{ for some } y\in \operatorname{Sm}(X)\cap S_Y\}.
\]
This is one of the clearest geometric characterizations of the notion [2407.14471, 2504.13464].

## 4. Finite-dimensional and sequence-space models

Classical sequence spaces display markedly different behavior. In \(\ell_1^n\), anti-coproximinal and strongly anti-coproximinal subspaces coincide. For a subspace \(Y=\operatorname{span}\{a_1,\dots,a_m\}\subset \ell_1^n\), the equivalent condition is that every component of the basis matrix satisfies the \(*\)-Property and
\[
|P_i^+\cup P_i^-|=1 \qquad \text{for all } i.
\]
Thus, in \(\ell_1^n\), the strong notion collapses to an exact combinatorial criterion [2407.14471].

In \(\ell_\infty^n\), the situation is opposite: there is no strongly anti-coproximinal subspace. For ordinary anti-coproximinality, the criterion depends on the zero set \(Z_A\) and the minimal norming set \(N\): if \(Z_A\neq\varnothing\), the subspace is not anti-coproximinal; if \(Z_A=\varnothing\), anti-coproximinality is equivalent to
\[
\dim(\operatorname{span}N)=n.
\]
The impossibility of the strong notion in \(\ell_\infty^n\) is a sharp contrast with \(\ell_1^n\) [2407.14471].

The spaces \(c_0\) and \(c\) admit a simple coordinatewise criterion. For a subspace \(Y\subset X\), where \(X=c_0\) or \(X=c\), the following are equivalent: \(Y\) is strongly anti-coproximinal; \(Y\) is anti-coproximinal; and for every \(r\in\mathbb N\) there exists \(y=(y_1,y_2,\dots)\in Y\) such that
\[
|y_r|>|y_n| \qquad \forall n\neq r.
\]
From this one obtains both positive and negative examples: \(c_0\) has no finite-dimensional anti-coproximinal subspaces, but it does admit infinite-dimensional strongly anti-coproximinal subspaces; \(c\) and \(\ell_\infty\) admit finite-dimensional strongly anti-coproximinal subspaces, including explicit two-dimensional examples generated by sine–cosine sequences [2504.13464].

Computational companions sharpen these finite-dimensional pictures. For subspaces of diagonal matrices \(\mathcal D_n\), equivalently of \(\ell_\infty^n\), coproximinality is characterized by the \(*\)-Property: an \(m\)-dimensional subspace is coproximinal iff there are exactly \(m\) nonequivalent components satisfying the \(*\)-Property, so non-coproximinality is equivalent to having more than \(m\) such components. Best coapproximants are determined by numerical-range constraints involving \(*\)-associated matrices [2407.20096]. In \(\ell_1^n\), the best coapproximation problem reduces to a finite linear system built from a minimal norming set; coproximinality is equivalent, in the zero-set-free case, to
\[
\dim(\operatorname{span}N)=\dim Y,
\]
and when the zero set is nonempty there is a threshold phenomenon along affine fibers: local neighborhoods may consist entirely of points with no best coapproximation [2407.20102]. These results do not formulate strong anti-coproximinality, but they supply explicit witness mechanisms for non-coproximinal and anti-coproximinal behavior.

## 5. Function spaces and operator spaces

In scalar-valued function spaces, anti-coproximinality and strong anti-coproximinality often coincide. For a proper closed subspace \(Y\subset \ell_\infty(K)\), the following are equivalent: \(Y\) is strongly anti-coproximinal; \(Y\) is anti-coproximinal; and for each \(k\in K\) there exists \(f\in Y\) such that
\[
|f(k)|>\lim_{n\to\infty}|f(k_n)|
\]
for every sequence \(\{k_n\}\subset K\) with \(k_n\neq k\) for all but finitely many \(n\). In particular, \(c_0\) is a strongly anti-coproximinal subspace of \(\ell_\infty\) [2504.13464].

For \(C_0(K)\), where \(K\) is locally connected, locally compact, and normal, a proper closed subspace \(Y\) is strongly anti-coproximinal iff it is anti-coproximinal, and this is equivalent to the local peak-set condition: for each nonempty open \(U\subset K\), there exists \(f\in Y\) such that
\[
M_f:=\{k\in K: |f(k)|=\|f\|\}\subset U.
\]
Under the additional assumptions that \(K\) is perfectly normal and has no isolated points, every finite-codimensional subspace of \(C_0(K)\) is strongly anti-coproximinal [2504.13464].

The vector-valued theory is subtler. Let \(\mathcal Y\subset C_0(K,\mathbb X)\), and for a selection map \(\psi\) choosing \(k_f\in M_f\) define
\[
\mathcal A_\psi=\{f(k_f): f\in S_{\mathcal Y}\}.
\]
If \(\mathcal Y\) is strongly anti-coproximinal, then \(\overline{\mathcal A_\psi}^{\,w}\) contains all w-ALUR points of \(B_{\mathbb X}\), \(\overline{\mathcal A_\psi}\) contains all ALUR points, and in finite-dimensional polyhedral \(\mathbb X\), \(\mathcal A_\psi\) intersects the interior of every maximal face of \(B_{\mathbb X}\). Under the dual-ball hypothesis
\[
B_{\mathbb X^*}= \overline{\operatorname{co}\big(w^*stExp(B_{\mathbb X^*})\big)}^{\,w^*},
\]
there is a complete characterization: \(\mathcal Y\) is strongly anti-coproximinal in \(C_0(K,\mathbb X)\) iff for every nonempty open set \(U\subset K\) and every nonempty weak-\(^*\)-open set \(V\subset S_{\mathbb X^*}\) containing a weak-\(^*\)-strongly exposed point, there exists \(f\in \mathcal Y\) such that
\[
M_f\subset U
\quad\text{and}\quad
J(f(k))\subset V \ \forall k\in M_f.
\]
In finite-dimensional real polyhedral \(\mathbb X\), this becomes equivalent to requiring norm-attaining values in the interior of arbitrary maximal faces [2508.04971].

Operator spaces supply another major source of examples. If \(B_X\) is the closed convex hull of its strongly exposed points and \(K(X,Y)\subsetneq L(X,Y)\), then
\[
K(X,Y)
\]
is strongly anti-coproximinal in
\[
L(X,Y).
\]
This applies, for example, when \(X\) has the Radon–Nikodým property. More generally, if \(Z\) is strongly anti-coproximinal in \(Y\), \(B_X=\overline{\operatorname{co}(stExp(B_X))}\), and \(\mathcal W\subset L(X,Y)\) contains the finite-rank operators \(\mathcal F(X,Z)\), then \(\mathcal W\) is strongly anti-coproximinal in \(L(X,Y)\). Under density of absolutely strongly exposing operators, this transfer becomes an equivalence between strong anti-coproximinality of \(Z\subset Y\) and that of suitable operator subspaces \(\mathcal W\subset L(X,Y)\) [2504.13464, 2508.04971].

## 6. Relation to coproximinality, proximinality, and broader approximation theory

Strong anti-coproximinality belongs to the negative side of best coapproximation theory. In generalized Minkowski spaces, positive coproximinality is already highly rigid: in dimension at least \(2\), every straight line is coproximinal iff the gauge is a norm, and in dimension at least \(3\), every closed codimension-one subspace is coproximinal iff the space is Hilbert. Moreover, if one finite-dimensional subspace is not coproximinal, there exists a closed hyperplane containing it such that every intermediate subspace inside that hyperplane is also not coproximinal. By contrapositive, these results identify structural settings in which non-coproximinal subspaces must exist and failure propagates upward in dimension [2101.05594].

A parallel but distinct line of work concerns proximinality rather than coproximinality. Read constructed an equivalent norm on \(c_0\) such that there are no proximinal subspaces of finite codimension \(n\ge 2\). This is an extreme negative finite-codimensional approximation phenomenon, but it is not formulated in coapproximation language [1307.7958]. Subsequent work on proximinal subspaces and norm-attaining functionals proved, for a non-reflexive Banach space \(X\) and a prescribed closed finite-codimensional subspace \(Y\) with codimension at least \(2\), that one can renorm \(X\) so that
\[
Y^\perp\subset NA(X)
\quad\text{and}\quad
Y \text{ is not proximinal},
\]
and even obtain antiproximinality via a quotient condition \(q(B_X)=U_{X/Y}\). That paper explicitly notes that the term “strongly anti-coproximinal” is absent, but its quotient-engineering constructions are natural raw material for stronger negative approximation notions [1503.06112].

These related theories clarify both the scope and the specificity of strongly anti-coproximinal subspaces. The coproximinal literature shows that ubiquitous coapproximation is rigid and often Hilbertian; the proximinal literature shows that finite-codimensional nearest-point phenomena can fail in extreme ways under renorming; the strong anti-coproximinal theory isolates the precise coapproximation-side analogue in terms of approximate Birkhoff–James orthogonality, support-function geometry, and facial structure of the unit ball. A plausible implication is that strongly anti-coproximinal subspaces should be viewed as boundary-saturating objects: in the strongest known characterizations, they must either realize every extreme support pattern of the ambient geometry or fail to exist altogether.

Source: https://www.emergentmind.com/topics/strongly-anti-coproximinal-subspaces