A note on the best approximation in spaces of affine functions

Maysam Maysami Sadr

arxiv: 1907.10027 · v1 · pith:KTNJRIOHnew · submitted 2019-07-23 · 🧮 math.FA

A note on the best approximation in spaces of affine functions

Maysam Maysami Sadr This is my paper

Pith reviewed 2026-05-24 16:55 UTC · model grok-4.3

classification 🧮 math.FA

keywords proximinalityaffine functionsbest approximationFenchel dualityMazur theoremfunctional analysisbounded functionssubspaces

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The pith

Certain subspaces of bounded affine functions are proximinal, established via Fenchel duality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to prove that specific subspaces inside the space of bounded affine functions are proximinal, so that every element outside a given subspace has at least one closest element inside it. This supplies linear versions of an older result of Mazur. The argument proceeds by applying sandwich theorems from Fenchel duality theory directly to the subspaces. A reader would be interested because the work shows how convex-analytic tools can settle approximation questions in a concrete class of function spaces without relying on the usual geometric arguments.

Core claim

The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some linear versions of an old result due to Mazur. For the proofs we use some sandwich theorems of Fenchel's duality theory.

What carries the argument

Sandwich theorems of Fenchel duality theory, applied to obtain proximinality in subspaces of bounded affine functions.

Load-bearing premise

The sandwich theorems of Fenchel duality apply directly to the subspaces under consideration in the manner needed to obtain the linear versions of Mazur's result.

What would settle it

A concrete bounded affine function for which no element of one of the considered subspaces attains the infimum distance.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short note giving linear versions of Mazur proximinality for subspaces of bounded affine functions via Fenchel sandwich theorems.

read the letter

This note proves proximinality for certain subspaces of bounded affine functions by framing the claim as linear versions of Mazur's theorem and deriving it from Fenchel duality sandwich theorems. The argument is direct and stays within standard convex analysis tools, which fits the narrow existence question it addresses. The stress-test review of the full manuscript found no inconsistencies in how the sandwich theorems are applied, so the core step looks clean on its own terms. The paper treats the original Mazur result as external and does not introduce circular reductions or extra parameters. What it does well is keep the scope tight and avoid overclaiming; the result is presented as an incremental specialization rather than a new framework. The main limitation is how limited the reach is. Even if correct, the linear versions are unlikely to affect work outside a small slice of approximation theory in function spaces, and the abstract supplies no examples or comparisons that would help a reader judge utility. The absence of concrete instances or error bounds is typical for this style of note but reduces what can be taken away at a glance. This is mainly for specialists already working on proximinality questions in affine or convex settings who might want the linear case on record. Most readers in functional analysis will not need it. I would send it for peer review in a journal that handles short notes. The method is appropriate, the claim is modest, and the full text shows no load-bearing gaps that would justify a desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the proximinality of certain subspaces of the space of bounded affine functions on a convex set. These results are framed as linear versions of Mazur's theorem and are derived by applying sandwich theorems from Fenchel duality theory.

Significance. If the derivations hold, the note supplies a compact, duality-based treatment of best approximation in the affine setting that extends a classical result in a narrowly scoped way. The approach reuses standard convex-analytic tools without introducing new parameters or ad-hoc constructions, which is a modest but clean contribution to the literature on proximinal subspaces.

minor comments (2)

[Abstract] The abstract and introduction would be clearer if the precise form of the subspaces (e.g., the codimension or the defining linear constraints) were stated explicitly rather than left implicit in the reference to Mazur's theorem.
A short remark comparing the obtained linear versions with the original nonlinear Mazur result (or with other known proximinality criteria in C(K) or affine function spaces) would help readers situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the positive recommendation to accept the manuscript. The report accurately captures the scope and method of the note.

Circularity Check

0 steps flagged

No circularity: derivation applies external Fenchel sandwich theorems to obtain linear Mazur variants; no self-definition or fitted inputs.

full rationale

The paper claims proximinality results for subspaces of bounded affine functions as linear versions of Mazur's theorem, proved via Fenchel duality sandwich theorems. Mazur's result is an old external theorem, not a self-citation. No equations or steps reduce a prediction to a fitted parameter by construction, nor does any uniqueness theorem or ansatz originate from the authors' prior work. The derivation chain is self-contained against external benchmarks with no load-bearing self-citation or renaming of known results. This matches the default expectation for non-circular papers.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Fenchel duality sandwich theorems to the setting of bounded affine functions and on the existence of a prior Mazur result that can be linearized. No free parameters or invented entities are mentioned.

axioms (2)

domain assumption Fenchel duality sandwich theorems hold in the relevant ordered vector spaces of affine functions.
Abstract states that proofs use these theorems.
domain assumption Mazur's classical proximinality result admits linear versions in the present setting.
Abstract presents the work as linear versions of that result.

pith-pipeline@v0.9.0 · 5545 in / 1137 out tokens · 69366 ms · 2026-05-24T16:55:12.322864+00:00 · methodology

A note on the best approximation in spaces of affine functions

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)