Anti-Coproximinal Subspaces in Banach Spaces

• Anti-coproximinal subspaces are defined by the complete absence of best coapproximations for any point outside them, with the strong variant also excluding any ε-best coapproximation.
• They are closely linked to Birkhoff–James orthogonality and supporting functionals, highlighting critical dual geometric structures and unit ball facial properties.
• Finite-dimensional and function/operator space models provide concrete criteria through dimension, facial intersections, and numerical-range conditions to characterize anti-coproximinality.

Anti-coproximinal subspaces are subspaces that exhibit systematic failure of best coapproximation. In the Banach-space setting of Sohel–Ghosh–Sain–Paul, a subspace $Y\subset X$ is anti-coproximinal if for any $x\in X\setminus Y$ there does not exist any best coapproximation to $x$ out of $Y$, and it is strongly anti-coproximinal if for any $x\in X\setminus Y$ and any $\varepsilon\in[0,1)$ there does not exist an $\varepsilon$-best coapproximation to $x$ out of $Y$ (Sohel et al., 2024). In the generalized Minkowski-space literature, the term can also be used in the weaker sense of mere failure of coproximinality (Jahn et al., 2021). Across these usages, the subject connects coapproximation to Birkhoff–James orthogonality, supporting functionals, weak$^*$-geometric structure in the dual, and the facial geometry of the unit ball (Ghosh et al., 7 Aug 2025).

1. Definitions and terminological scope

Let $x\in X\setminus Y$0 be a real Banach space and $x\in X\setminus Y$1 a subspace. An element $x\in X\setminus Y$2 is a best coapproximation to $x\in X\setminus Y$3 out of $x\in X\setminus Y$4 if

$x\in X\setminus Y$5

The set of all best coapproximations to $x\in X\setminus Y$6 from $x\in X\setminus Y$7 is denoted $x\in X\setminus Y$8. A subspace is coproximinal if $x\in X\setminus Y$9 for every $x$0, and it is co-Chebyshev if it is coproximinal and $x$1 is a singleton for every $x$2. The approximate variant replaces exact coapproximation by $x$3-best coapproximation, defined through approximate Birkhoff–James orthogonality (Sohel et al., 2024).

In this Banach-space framework, anti-coproximinal means

$x$4

while strongly anti-coproximinal means that for every $x$5 and every $x$6, there is no $x$7-best coapproximation to $x$8 out of $x$9. Every strongly anti-coproximinal subspace is anti-coproximinal, but the converse fails in general; explicit counterexamples are given in finite-dimensional settings (Sohel et al., 2024).

A persistent terminological issue is that the generalized Minkowski-space paper uses anti-coproximinal in the broader sense of “not coproximinal,” namely the existence of some point $Y$0 for which the best coapproximation set is empty. That broader usage is natural in gauge geometry, but it is strictly weaker than the Banach-space definition requiring failure for every point outside the subspace. This terminological divergence is one of the main points to keep in view when comparing results across the literature (Jahn et al., 2021).

2. Orthogonality and dual formulations

The main analytic mechanism behind anti-coproximinality is Birkhoff–James orthogonality. For $Y$1,

$Y$2

Given a subspace $Y$3 and $Y$4, $Y$5 is a best coapproximation to $Y$6 out of $Y$7 if and only if

$Y$8

that is, $Y$9 for all $x\in X\setminus Y$0. For $x\in X\setminus Y$1, the paper uses Chmieliński’s approximate orthogonality: $x\in X\setminus Y$2 and proves that $x\in X\setminus Y$3 is equivalent to the existence of $x\in X\setminus Y$4 such that $x\in X\setminus Y$5, where

$x\in X\setminus Y$6

is the set of supporting functionals at $x\in X\setminus Y$7 (Sohel et al., 2024).

This orthogonality viewpoint admits a dual-geometric reformulation through selection maps. For a subspace $x\in X\setminus Y$8, a selection map is a map $x\in X\setminus Y$9 with $\varepsilon\in[0,1)$0 for every $\varepsilon\in[0,1)$1, together with the natural homogeneity condition in the complex case. The general characterization states that $\varepsilon\in[0,1)$2 is anti-coproximinal if and only if

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

Under the hypothesis

$\varepsilon\in[0,1)$4

a closed proper subspace $\varepsilon\in[0,1)$5 is strongly anti-coproximinal if and only if

$\varepsilon\in[0,1)$6

The same framework yields

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

together with corresponding criteria for coproximinality and co-Chebyshevness (Ghosh et al., 7 Aug 2025).

In smooth Banach spaces these criteria simplify because each $\varepsilon\in[0,1)$8 is a singleton. If

$\varepsilon\in[0,1)$9

then $\varepsilon$0 is anti-coproximinal if and only if $\varepsilon$1; in finite-dimensional smooth spaces this becomes the dimension condition $\varepsilon$2 (Sohel et al., 2024).

3. Dimension-dependent structure and rigidity

A central theme is that coproximinality in low codimension imposes strong geometric restrictions on the ambient space. In generalized Minkowski spaces $\varepsilon$3, Theorem 3.2 shows that for $\varepsilon$4, every straight line is coproximinal if and only if $\varepsilon$5 is a norm. The contrapositive yields a robust existence theorem: if $\varepsilon$6 is not a norm, then some line is anti-coproximinal in the broad generalized-Minkowski sense. The argument is already visible in dimension $\varepsilon$7, where non-symmetry of the unit ball produces a line $\varepsilon$8 with $\varepsilon$9 for a suitable point $x$0 (Jahn et al., 2021).

In dimension at least $x$1, the rigidity becomes much sharper. Theorem 5.1 states that every closed $x$2-codimensional linear subspace is coproximinal if and only if the space is a Hilbert space, meaning that the gauge is induced by an inner product and the space is complete. The proof splits into two parts. First, if every closed hyperplane is coproximinal, then the gauge must be induced by an inner product; second, if the norm comes from an inner product, coproximinality of every closed hyperplane is equivalent to completeness through the Riesz representation theorem. Consequently, in any non-Hilbert generalized Minkowski space of dimension $x$3, or in any incomplete inner product space, anti-coproximinal closed hyperplanes must exist (Jahn et al., 2021).

The same codimension-$x$4 rigidity reappears in smooth Banach spaces. If $x$5 is smooth and $x$6, then $x$7 is Hilbert if and only if there is no anti-coproximinal closed hyperplane in $x$8. For $x$9, $Y$0 and $Y$1, this becomes the concrete characterization

$Y$2

These statements place anti-coproximinal hyperplanes alongside classical Hilbert-space characterizations by orthogonality and projection properties (Sohel et al., 2024).

Dimension also governs propagation phenomena. Proposition 5.2 in the generalized Minkowski-space setting states that if there exists a finite-dimensional linear subspace $Y$3 that is not coproximinal, then there exists a closed $Y$4-codimensional subspace $Y$5 such that $Y$6 is not coproximinal and every intermediate subspace $Y$7 with $Y$8 is also not coproximinal. Conversely, if all subspaces of some fixed finite dimension are coproximinal, then all lower finite-dimensional subspaces are coproximinal as well. This establishes a precise dimension-dependent monotonicity of coapproximation failure and success (Jahn et al., 2021).

4. Geometric criteria and obstructions in Banach spaces

Strong anti-coproximinality is compatible only with rather singular geometry. A general sufficient condition is the following: if for each $Y$9 there exists $^*$0 such that

$^*$1

then $^*$2 is strongly anti-coproximinal. The intuition is that the supporting functionals of such a $^*$3 are so tightly aligned with those of $^*$4 and $^*$5 that $^*$6-orthogonality cannot occur. In the opposite direction, if $^*$7 is reflexive, $^*$8 has the Kadets–Klee property, and $^*$9 is a closed strongly anti-coproximinal subspace, then for each $x\in X\setminus Y$00 there exists $x\in X\setminus Y$01 with

$x\in X\setminus Y$02

This necessary condition immediately rules out strongly anti-coproximinal closed subspaces in many familiar classes: reflexive strictly convex spaces whose dual has Kadets–Klee, reflexive smooth spaces whose dual has Kadets–Klee, finite-dimensional smooth spaces, finite-dimensional strictly convex spaces, and uniformly smooth spaces (Sohel et al., 2024).

The function-space study sharpens these obstructions in terms of points and faces of the unit ball. If $x\in X\setminus Y$03 is a closed proper strongly anti-coproximinal subspace of $x\in X\setminus Y$04, then every w-ALUR point of $x\in X\setminus Y$05 must belong to $x\in X\setminus Y$06. In finite dimension, a strongly anti-coproximinal subspace must intersect every maximal face of $x\in X\setminus Y$07. Moreover, if a subspace intersects the relative interior of every facet of $x\in X\setminus Y$08, then it is strongly anti-coproximinal. In finite-dimensional polyhedral spaces this face-intersection condition becomes an equivalence, yielding a purely facial characterization of strong anti-coproximinality (Sohel et al., 18 Apr 2025).

These results also dispel a common misconception: intersecting every maximal face is necessary for finite-dimensional strong anti-coproximinality, but it is not sufficient in general. The literature contains examples where a subspace intersects every maximal face and is nevertheless coproximinal rather than anti-coproximinal. The correct sufficient condition is stronger, namely intersection of the relative interior of every facet in the polyhedral setting (Sohel et al., 18 Apr 2025).

5. Explicit classifications in finite-dimensional model spaces

The recent literature provides unusually concrete descriptions of anti-coproximinal behavior in finite-dimensional spaces. In finite-dimensional polyhedral Banach spaces, if $x\in X\setminus Y$09 is dense in $x\in X\setminus Y$10, then $x\in X\setminus Y$11 is anti-coproximinal if and only if

$x\in X\setminus Y$12

For strong anti-coproximinality there is an even more geometric criterion: $x\in X\setminus Y$13 is strongly anti-coproximinal if and only if $x\in X\setminus Y$14 intersects the interior of every facet of $x\in X\setminus Y$15, equivalently,

$x\in X\setminus Y$16

The same paper gives examples showing that anti-coproximinal and strongly anti-coproximinal need not coincide, and proves that in $x\in X\setminus Y$17 they do coincide for proper subspaces, with a combinatorial characterization through the $x\in X\setminus Y$18-property and the condition $x\in X\setminus Y$19 for all components $x\in X\setminus Y$20 (Sohel et al., 2024).

A complementary line of work gives a complete computational treatment of best coapproximation in $x\in X\setminus Y$21. When the zero set $x\in X\setminus Y$22 is empty, the subspace $x\in X\setminus Y$23 has a unique minimal norming set $x\in X\setminus Y$24, and if $x\in X\setminus Y$25 and $x\in X\setminus Y$26, then

$x\in X\setminus Y$27

If $x\in X\setminus Y$28, then $x\in X\setminus Y$29 is coproximinal if and only if its reduced subspace $x\in X\setminus Y$30 is coproximinal in the lower-dimensional $x\in X\setminus Y$31, while no subspace with nonempty zero set is co-Chebyshev. The best coapproximation problem is reduced to solvability of a finite linear system, so non-coproximinality becomes an explicit inconsistency phenomenon. This supplies a tractable finite-dimensional mechanism for producing failure of coproximinality in the broader sense often associated with anti-coproximinal behavior (Sain et al., 2024).

For subspaces of diagonal matrices and, equivalently, of $x\in X\setminus Y$32, the $x\in X\setminus Y$33-property again governs the theory. If $x\in X\setminus Y$34, and $x\in X\setminus Y$35 is the number of nonequivalent components satisfying the $x\in X\setminus Y$36-property, then

$x\in X\setminus Y$37

while $x\in X\setminus Y$38 is co-Chebyshev if and only if $x\in X\setminus Y$39 and each relevant equivalence class $x\in X\setminus Y$40 has size $x\in X\setminus Y$41. Best coapproximations are characterized by numerical-range constraints involving $x\in X\setminus Y$42-associated matrices, so non-coproximinality appears as failure of solvability of a finite system of numerical-range conditions. Through the natural isometric identification of $x\in X\setminus Y$43 with $x\in X\setminus Y$44, this yields a complete diagonal-matrix model for finite-dimensional coapproximation failure (Sain et al., 2024).

6. Function spaces and operator spaces

In scalar function spaces, anti-coproximinal and strongly anti-coproximinal behavior often coincide and can be detected by peaking conditions. For a proper closed subspace $x\in X\setminus Y$45, the following are equivalent: $x\in X\setminus Y$46 is strongly anti-coproximinal, $x\in X\setminus Y$47 is anti-coproximinal, and for each $x\in X\setminus Y$48 there exists $x\in X\setminus Y$49 such that $x\in X\setminus Y$50 strictly dominates the limsup of $x\in X\setminus Y$51 along every sequence $x\in X\setminus Y$52 eventually. In $x\in X\setminus Y$53 and $x\in X\setminus Y$54, the criterion simplifies to the existence, for each coordinate $x\in X\setminus Y$55, of a vector $x\in X\setminus Y$56 with $x\in X\setminus Y$57 for all $x\in X\setminus Y$58. As consequences, $x\in X\setminus Y$59 has no finite-dimensional anti-coproximinal subspace, whereas infinite-dimensional strongly anti-coproximinal subspaces do exist; in $x\in X\setminus Y$60 and $x\in X\setminus Y$61, even finite-dimensional strongly anti-coproximinal examples occur (Sohel et al., 18 Apr 2025).

For scalar $x\in X\setminus Y$62, the topological geometry of $x\in X\setminus Y$63 becomes decisive. If $x\in X\setminus Y$64 is locally compact normal and $x\in X\setminus Y$65 is anti-coproximinal, then for every nonempty open set $x\in X\setminus Y$66 there exists $x\in X\setminus Y$67 with norm-attainment set $x\in X\setminus Y$68. If $x\in X\setminus Y$69 is locally connected, locally compact, and normal, then for closed proper $x\in X\setminus Y$70 the following are equivalent: $x\in X\setminus Y$71 is strongly anti-coproximinal, $x\in X\setminus Y$72 is anti-coproximinal, and for every nonempty open $x\in X\setminus Y$73 there exists $x\in X\setminus Y$74 with $x\in X\setminus Y$75. When $x\in X\setminus Y$76 is locally connected, locally compact, perfectly normal, and has no isolated points, every finite-codimensional subspace of $x\in X\setminus Y$77 is strongly anti-coproximinal (Sohel et al., 18 Apr 2025).

The vector-valued theory in $x\in X\setminus Y$78 is more delicate and uses weak$x\in X\setminus Y$79-strongly exposed points of $x\in X\setminus Y$80. Under the assumptions that $x\in X\setminus Y$81 is locally compact normal and

$x\in X\setminus Y$82

a closed subspace $x\in X\setminus Y$83 is strongly anti-coproximinal if and only if for every nonempty open $x\in X\setminus Y$84 and every nonempty weak$x\in X\setminus Y$85-open set $x\in X\setminus Y$86 containing a weak$x\in X\setminus Y$87-strongly exposed point of $x\in X\setminus Y$88, there exists $x\in X\setminus Y$89 such that

$x\in X\setminus Y$90

In finite-dimensional real polyhedral $x\in X\setminus Y$91, this simplifies to a face condition: $x\in X\setminus Y$92 is strongly anti-coproximinal if and only if for each open $x\in X\setminus Y$93 and each maximal face $x\in X\setminus Y$94 of $x\in X\setminus Y$95, there exists $x\in X\setminus Y$96 with $x\in X\setminus Y$97 and $x\in X\setminus Y$98 for all $x\in X\setminus Y$99 (Ghosh et al., 7 Aug 2025).

Operator spaces admit both structural characterizations and stability theorems. If the unit ball $x$00 is the closed convex hull of its strongly exposed points and $x$01, then $x$02 is strongly anti-coproximinal in $x$03. In particular, if $x$04 has the Radon–Nikodým property, then either all bounded operators are compact or compact operators form a strongly anti-coproximinal subspace of the full operator space (Sohel et al., 18 Apr 2025).

A more general operator-space stability principle is available. If $x$05 separates $x$06, $x$07, and $x$08 is dense in $x$09, then anti-coproximinality of $x$10 is equivalent to anti-coproximinality of $x$11, and the same holds for strong anti-coproximinality under the corresponding stronger geometric assumption. In particular, if $x$12 has the Radon–Nikodým property and $x$13, then

$x$14

A concrete example given in the paper is that $x$15 is strongly anti-coproximinal in $x$16, hence $x$17 is strongly anti-coproximinal in $x$18 for $x$19 (Ghosh et al., 7 Aug 2025).

7. Relation to proximinality and broader negative geometry

Anti-coproximinality belongs to the “best coapproximation” side of Banach-space geometry, but it has a close conceptual analogue on the “best approximation” side. Read’s renorming of $x$20 furnishes a Banach space with no proximinal subspace of finite codimension $x$21, even though proximinal hyperplanes remain dense by the Bishop–Phelps–Bollobás theorem. The result is not about coproximinality, but it shows that codimension-$x$22 positivity may coexist with extreme failure in higher codimension (Read, 2013).

This suggests a useful caution for the coapproximation theory. Results such as “all lines are coproximinal” or “all closed hyperplanes are coproximinal” are exceptionally rigid and force norm or Hilbert structure, but they do not justify extrapolation to more general subspaces. The existing anti-coproximinal theory repeatedly exhibits the same pattern: codimension, facial geometry, and dual support structure control the passage from existence to systematic nonexistence of best coapproximations (Jahn et al., 2021).

The modern theory therefore presents anti-coproximinal subspaces as geometric detectors of failure. In smooth spaces they are governed by the weak$x$23-span of support functionals; in polyhedral spaces by the way a subspace meets facets; in function spaces by the ability to peak on arbitrarily small open sets; and in operator spaces by the availability of rank-one or absolutely strongly exposing test operators. What varies across these settings is the technical language, but the underlying phenomenon is stable: anti-coproximinality marks the absence of global orthogonality-compatible coapproximation schemes, while strong anti-coproximinality excludes even approximate versions of such schemes (Sohel et al., 2024).