Strongly Anti-Coproximinal Subspaces

• Strongly anti-coproximinal subspaces are defined as those in which, for every x outside the subspace and every ε in [0,1), no ε-best coapproximation exists.
• The concept is characterized through approximate Birkhoff–James orthogonality, linking support functionals and the facial structure of the unit ball with duality conditions.
• Examples in various Banach spaces, including ℓ₁ and ℓ∞, illustrate how these subspaces reveal intricacies in non-coproximinality and approximation theory.

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 (Sohel et al., 2024, Ghosh et al., 7 Aug 2025).

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$0 out of $X$1 if

$X$2

The set of all best coapproximations is denoted $X$3. In contrast with best approximation, $X$4 may be empty even in finite-dimensional settings (Sohel et al., 2024).

For $X$5, $X$6 is an $X$7-best coapproximation to $X$8 out of $X$9 if

$x\in X\setminus Y$0

that is, $x\in X\setminus Y$1 for every $x\in X\setminus Y$2. Here approximate Birkhoff–James orthogonality is given by

$x\in X\setminus Y$3

A subspace $x\in X\setminus Y$4 is coproximinal if $x\in X\setminus Y$5 for every $x\in X\setminus Y$6. It is anti-coproximinal if

$x\in X\setminus Y$7

It is strongly anti-coproximinal if for every $x\in X\setminus Y$8 and every $x\in X\setminus Y$9, there is no $\varepsilon\in[0,1)$0-best coapproximation to $\varepsilon\in[0,1)$1 out of $\varepsilon\in[0,1)$2. Thus

$\varepsilon\in[0,1)$3

but not conversely (Sohel et al., 2024).

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 $\varepsilon\in[0,1)$4-based examples discussed in the literature (Sohel et al., 2024).

2. Orthogonality and duality formulations

The central structural identity is that best coapproximation is an orthogonality condition: $\varepsilon\in[0,1)$5 Accordingly, strong anti-coproximinality can be read as the global failure of approximate orthogonality: $\varepsilon\in[0,1)$6 This places the subject squarely in the geometry of Birkhoff–James orthogonality (Sohel et al., 2024).

If

$\varepsilon\in[0,1)$7

then the standard dual criterion for approximate orthogonality is

$\varepsilon\in[0,1)$8

For $\varepsilon\in[0,1)$9-best coapproximation, this yields the equivalent condition: $\varepsilon$0 is an $\varepsilon$1-best coapproximation to $\varepsilon$2 iff for every $\varepsilon$3, there exists $\varepsilon$4 such that

$\varepsilon$5

At $\varepsilon$6, one recovers the exact coapproximation criterion (Sohel et al., 2024).

A more abstract dual description uses selection maps. For a subspace $\varepsilon$7, a selection map is a map

$\varepsilon$8

with the natural homogeneity condition $\varepsilon$9. Then anti-coproximinality is equivalent to

$x$0

Under the dual-ball hypothesis

$x$1

a closed proper subspace $x$2 is strongly anti-coproximinal iff for every selection map $x$3,

$x$4

This shows that anti-coproximinality is a weak-$x$5 spanning condition, whereas strong anti-coproximinality requires approximation of the weak-$x$6-strongly exposed boundary of the dual ball by support functionals coming from the subspace (Ghosh et al., 7 Aug 2025).

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$7 there exists $x$8 such that

$x$9

then $Y$0 is strongly anti-coproximinal. The condition requires a strong alignment of support functionals of $Y$1 with those of $Y$2 or $Y$3, and it rules out all approximate coapproximants with $Y$4 (Sohel et al., 2024).

A general necessary condition appears in the reflexive/Kadets–Klee setting. If $Y$5 is reflexive, $Y$6 has the Kadets–Klee property, $Y$7 is closed, and $Y$8 is strongly anti-coproximinal, then for each $Y$9 there exists $X$0 such that

$X$1

This necessary condition is not sufficient; explicit counterexamples show that shared support functionals do not by themselves force strong anti-coproximinality (Sohel et al., 2024).

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$2 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 (Sohel et al., 2024).

The geometry of the ambient unit ball yields further necessary conditions. If $X$3 is a closed proper strongly anti-coproximinal subspace, then $X$4 must contain all w-ALUR points of $X$5. For finite-dimensional $X$6, strong anti-coproximinality implies that $X$7 intersects every maximal face of $X$8. A stronger sufficient condition is also known: if $X$9 intersects the relative interior of every facet of $Y\subset X$0, then $Y\subset X$1 is strongly anti-coproximinal. In finite-dimensional polyhedral Banach spaces these conditions become exact: $Y\subset X$2 is strongly anti-coproximinal iff $Y\subset X$3 intersects the interior of every facet of $Y\subset X$4, equivalently iff

$Y\subset X$5

where

$Y\subset X$6

This is one of the clearest geometric characterizations of the notion (Sohel et al., 2024, Sohel et al., 18 Apr 2025).

4. Finite-dimensional and sequence-space models

Classical sequence spaces display markedly different behavior. In $Y\subset X$7, anti-coproximinal and strongly anti-coproximinal subspaces coincide. For a subspace $Y\subset X$8, the equivalent condition is that every component of the basis matrix satisfies the $Y\subset X$9-Property and

$y_0\in Y$0

Thus, in $y_0\in Y$1, the strong notion collapses to an exact combinatorial criterion (Sohel et al., 2024).

In $y_0\in Y$2, the situation is opposite: there is no strongly anti-coproximinal subspace. For ordinary anti-coproximinality, the criterion depends on the zero set $y_0\in Y$3 and the minimal norming set $y_0\in Y$4: if $y_0\in Y$5, the subspace is not anti-coproximinal; if $y_0\in Y$6, anti-coproximinality is equivalent to

$y_0\in Y$7

The impossibility of the strong notion in $y_0\in Y$8 is a sharp contrast with $y_0\in Y$9 (Sohel et al., 2024).

The spaces $X$00 and $X$01 admit a simple coordinatewise criterion. For a subspace $X$02, where $X$03 or $X$04, the following are equivalent: $X$05 is strongly anti-coproximinal; $X$06 is anti-coproximinal; and for every $X$07 there exists $X$08 such that

$X$09

From this one obtains both positive and negative examples: $X$10 has no finite-dimensional anti-coproximinal subspaces, but it does admit infinite-dimensional strongly anti-coproximinal subspaces; $X$11 and $X$12 admit finite-dimensional strongly anti-coproximinal subspaces, including explicit two-dimensional examples generated by sine–cosine sequences (Sohel et al., 18 Apr 2025).

Computational companions sharpen these finite-dimensional pictures. For subspaces of diagonal matrices $X$13, equivalently of $X$14, coproximinality is characterized by the $X$15-Property: an $X$16-dimensional subspace is coproximinal iff there are exactly $X$17 nonequivalent components satisfying the $X$18-Property, so non-coproximinality is equivalent to having more than $X$19 such components. Best coapproximants are determined by numerical-range constraints involving $X$20-associated matrices (Sain et al., 2024). In $X$21, 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

$X$22

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 (Sain et al., 2024). 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 $X$23, the following are equivalent: $X$24 is strongly anti-coproximinal; $X$25 is anti-coproximinal; and for each $X$26 there exists $X$27 such that

$X$28

for every sequence $X$29 with $X$30 for all but finitely many $X$31. In particular, $X$32 is a strongly anti-coproximinal subspace of $X$33 (Sohel et al., 18 Apr 2025).

For $X$34, where $X$35 is locally connected, locally compact, and normal, a proper closed subspace $X$36 is strongly anti-coproximinal iff it is anti-coproximinal, and this is equivalent to the local peak-set condition: for each nonempty open $X$37, there exists $X$38 such that

$X$39

Under the additional assumptions that $X$40 is perfectly normal and has no isolated points, every finite-codimensional subspace of $X$41 is strongly anti-coproximinal (Sohel et al., 18 Apr 2025).

The vector-valued theory is subtler. Let $X$42, and for a selection map $X$43 choosing $X$44 define

$X$45

If $X$46 is strongly anti-coproximinal, then $X$47 contains all w-ALUR points of $X$48, $X$49 contains all ALUR points, and in finite-dimensional polyhedral $X$50, $X$51 intersects the interior of every maximal face of $X$52. Under the dual-ball hypothesis

$X$53

there is a complete characterization: $X$54 is strongly anti-coproximinal in $X$55 iff for every nonempty open set $X$56 and every nonempty weak-$X$57-open set $X$58 containing a weak-$X$59-strongly exposed point, there exists $X$60 such that

$X$61

In finite-dimensional real polyhedral $X$62, this becomes equivalent to requiring norm-attaining values in the interior of arbitrary maximal faces (Ghosh et al., 7 Aug 2025).

Operator spaces supply another major source of examples. If $X$63 is the closed convex hull of its strongly exposed points and $X$64, then

$X$65

is strongly anti-coproximinal in

$X$66

This applies, for example, when $X$67 has the Radon–Nikodým property. More generally, if $X$68 is strongly anti-coproximinal in $X$69, $X$70, and $X$71 contains the finite-rank operators $X$72, then $X$73 is strongly anti-coproximinal in $X$74. Under density of absolutely strongly exposing operators, this transfer becomes an equivalence between strong anti-coproximinality of $X$75 and that of suitable operator subspaces $X$76 (Sohel et al., 18 Apr 2025, Ghosh et al., 7 Aug 2025).

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 $X$77, every straight line is coproximinal iff the gauge is a norm, and in dimension at least $X$78, 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 (Jahn et al., 2021).

A parallel but distinct line of work concerns proximinality rather than coproximinality. Read constructed an equivalent norm on $X$79 such that there are no proximinal subspaces of finite codimension $X$80. This is an extreme negative finite-codimensional approximation phenomenon, but it is not formulated in coapproximation language (Read, 2013). Subsequent work on proximinal subspaces and norm-attaining functionals proved, for a non-reflexive Banach space $X$81 and a prescribed closed finite-codimensional subspace $X$82 with codimension at least $X$83, that one can renorm $X$84 so that

$X$85

and even obtain antiproximinality via a quotient condition $X$86. 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 (Rmoutil, 2015).

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.