arxiv: 2605.03797 · v1 · submitted 2026-05-05 · 🧮 math.FA · math.MG· math.OC

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Generalized outer linearizations and extremal properties of rotational epi-symmetrizations

Steven Hoehner , Fabian Mussnig

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classification 🧮 math.FA math.MGmath.OC

keywords generalized outer linearizationsrotational epi-symmetrizationsextremal principlesconvex functionsepi-convergenceUrysohn inequalitypiecewise affine approximations

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The pith

The rotational epi-symmetrization maximizes best approximations under generalized outer linearizations for monotone concave functionals on coercive convex functions.

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

The paper extends a classical extremal principle for convex bodies to the functional setting by defining generalized outer linearizations as convex minorants built from function-specific sets of slopes. It establishes that the rotational epi-symmetrization of a coercive convex function delivers the maximal value among all such linearizations when evaluated by any monotone concave functional that is upper semicontinuous with respect to epi-convergence. This framework operates without requiring super-coercivity, using duality and epi-convergence tools to handle the resulting challenges. The findings produce a functional analogue of Urysohn's inequality and extend older results on coverings and approximations in convex geometry.

Core claim

On a standard class of coercive convex functions, the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence.

What carries the argument

Generalized outer linearization, a convex minorant of the function represented by a function-dependent set of slopes that converts geometric supporting halfspaces into supporting affine functions via the Legendre-Fenchel transform.

If this is right

A functional version of Urysohn's inequality follows directly from the extremal principle.
An analytic extension of the classical covering result by Firey and Groemer is obtained.
An extremal inequality holds for the piecewise affine approximation of convex functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The principle may apply to bounding errors in numerical convex optimization using symmetrized approximations.
Similar extremal results could hold for other types of symmetrizations or in non-Euclidean settings.
Refined duality arguments developed here might resolve variational issues in related problems involving epi-convergence.

Load-bearing premise

The functionals to which the extremal principle applies are assumed to be monotone, concave, and upper semicontinuous with respect to epi-convergence.

What would settle it

Construct a coercive convex function whose rotational epi-symmetrization gives a strictly lower value than some other outer linearization under a specific monotone concave functional that is epi-convergence upper semicontinuous.

Figures

Figures reproduced from arXiv: 2605.03797 by Fabian Mussnig, Steven Hoehner.

**Figure 1.** Figure 1: On the left, an outer linearization of ψ(x) = x 2 with four slopes is shown in blue. On the right, the corresponding outer log-linearization of f(x) = e −x 2 is shown in red view at source ↗

read the original abstract

We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer linearization is defined as a convex minorant represented by a general but function-dependent set of slopes, thereby extending classical outer representations of convex bodies by supporting halfspaces. This representation converts geometric outer approximations by supporting halfspaces into functional approximations by supporting affine functions, and replaces outer normal data by a dual sampling problem in the domain of the Legendre--Fenchel transform. On a standard class of coercive convex functions, we derive a general extremal principle, showing that the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence. A central feature of the analysis is that it is carried out in the natural class of coercive, but not necessarily super-coercive, convex functions. Working in this setting introduces intricate topological and variational difficulties, which are addressed using refined duality and epi-convergence arguments. As an application of our main results, we derive a functional version of Urysohn's inequality, as well as an analytic extension of a classical covering result of Firey and Groemer (J. London Math. Soc., 1964). Finally, we prove an extremal inequality related to the piecewise affine approximation of convex functions.

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

This paper defines generalized outer linearizations to extend Schneider's 1967 extremal principle to coercive convex functions and proves that rotational epi-symmetrizations maximize approximations for a broad class of monotone concave functionals.

read the letter

The main advance is the new notion of generalized outer linearizations: convex minorants built from function-dependent slopes, turned into a dual sampling problem via the Legendre-Fenchel transform. This lets the authors carry Schneider's geometric statement over to the functional setting and show that rotational epi-symmetrization is extremal for any monotone concave functional that is upper semicontinuous with respect to epi-convergence. They stay inside the natural class of coercive (not necessarily super-coercive) convex functions, which forces them to use refined duality and epi-convergence arguments to control the topology. That part looks like the real technical work. They also recover a functional Urysohn inequality and an analytic version of the Firey-Groemer covering result, plus an extremal statement for piecewise-affine approximations. Those applications are concrete and follow directly from the main theorem. The abstract and stress-test give no sign of circularity or hidden fitting; the cited 1967 and 1964 papers are treated as starting points rather than re-derived. The only soft spot visible from the given material is that the topological and variational difficulties are described as intricate but not illustrated with a short explicit example or counter-example check in the abstract itself. If the full proofs close those gaps cleanly, the contribution is solid for the convex-analysis literature. This is the kind of paper that belongs in a specialized journal like Journal of Convex Analysis or Advances in Mathematics. A serious editor should send it to referees who know epi-convergence and symmetrization inequalities; the claims are specific enough to be checked and the extension is not routine.

Referee Report

0 major / 4 minor

Summary. The manuscript develops a functional extension of Schneider's 1967 extremal principle by introducing generalized outer linearizations of coercive convex functions on R^n. These are defined as convex minorants with function-dependent sets of slopes, converting geometric outer approximations by supporting halfspaces into functional approximations via the Legendre-Fenchel transform. The central result is an extremal principle asserting that the rotational epi-symmetrization maximizes best approximations under outer linearizations for any monotone concave functional that is upper semicontinuous with respect to epi-convergence. The analysis is performed in the class of coercive (not necessarily super-coercive) convex functions using refined duality and epi-convergence arguments. Applications include a functional version of Urysohn's inequality, an analytic extension of the Firey-Groemer covering result, and an extremal inequality for piecewise affine approximations.

Significance. If the derivations hold, the work provides a meaningful bridge between classical convex geometry and modern functional analysis by extending geometric symmetrization principles to coercive convex functions. Credit is due for addressing the topological difficulties of the coercive setting without restricting to super-coercive cases, for the parameter-free character of the extremal property, and for deriving concrete applications such as the functional Urysohn inequality. This could support further developments in variational analysis and approximation theory.

minor comments (4)

The introduction would benefit from a short roadmap outlining the key steps in the duality and epi-convergence arguments used to handle the coercive case.
Notation for the generalized outer linearization (e.g., the precise role of the function-dependent slope set) should be introduced with a displayed definition or diagram for clarity.
In the applications section, explicitly verify that the chosen monotone concave functionals satisfy upper semicontinuity with respect to epi-convergence to make the reductions self-contained.
Ensure consistent use of terminology such as 'rotational epi-symmetrization' across the abstract, introduction, and main statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. The recognition of the extension of Schneider's principle, the handling of the coercive (not necessarily super-coercive) setting, and the derived applications is appreciated.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces generalized outer linearizations as a new functional extension of supporting halfspaces for coercive convex functions, then derives the extremal property of rotational epi-symmetrizations for monotone concave functionals via duality and epi-convergence. This rests on external citations to Schneider (1967) and Firey-Groemer (1964) as independent starting points rather than self-citations. No definitions, equations, or steps in the abstract or described results reduce the claimed extremal principle to a fitted input, renamed known result, or load-bearing self-citation chain; the analysis adds independent variational content for the non-super-coercive setting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on abstract only; the central claim rests on standard background from convex analysis together with the new definition of generalized outer linearizations.

axioms (1)

domain assumption Coercive convex functions on R^n admit well-behaved Legendre-Fenchel transforms and epi-convergence properties
The paper explicitly works in the class of coercive (but not necessarily super-coercive) convex functions and invokes refined duality and epi-convergence arguments.

invented entities (1)

generalized outer linearization no independent evidence
purpose: Convex minorant represented by a function-dependent set of slopes, converting geometric outer approximations into functional ones
Introduced in the abstract as the central new representation tool extending supporting halfspaces.

pith-pipeline@v0.9.0 · 5570 in / 1385 out tokens · 65126 ms · 2026-05-07T13:04:09.605014+00:00 · methodology

discussion (0)

Reference graph

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