On the Strong Quasiconvexity of Norms and Distance Functions

N. M. Nam; V. S. T. Long

arxiv: 2605.25614 · v1 · pith:7B26V3ESnew · submitted 2026-05-25 · 🧮 math.OC

On the Strong Quasiconvexity of Norms and Distance Functions

V. S. T. Long , N. M. Nam This is my paper

Pith reviewed 2026-06-29 20:54 UTC · model grok-4.3

classification 🧮 math.OC

keywords strong quasiconvexitynorm functionsdistance functionsfinite-dimensional spacesconvex setsquasiconvex optimization

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The pith

Necessary and sufficient conditions are established for norms to be strongly quasiconvex on convex sets.

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

The paper aims to characterize when a norm function is strongly quasiconvex on a convex set by providing necessary and sufficient conditions in finite-dimensional spaces. It also starts investigating strong quasiconvexity for distance functions. This matters because it generalizes the known strong quasiconvexity of the Euclidean norm on bounded sets to arbitrary norms, aiding understanding of their optimization properties. The results offer geometric insights and build on existing literature.

Core claim

We establish necessary and sufficient conditions for a norm function to be strongly quasiconvex on a convex set. We also initiate the study of the strong quasiconvexity of distance functions. Our results provide new insights into the geometric properties of norm and distance functions and extend several existing results in the literature.

What carries the argument

the necessary and sufficient geometric conditions for the strong quasiconvexity of norm functions on convex sets

Load-bearing premise

The underlying space is finite-dimensional.

What would settle it

A finite-dimensional norm that satisfies the stated necessary and sufficient conditions but fails to be strongly quasiconvex on some convex set, or the converse.

read the original abstract

This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this property for general norms remains open. We establish necessary and sufficient conditions for a norm function to be strongly quasiconvex on a convex set. We also initiate the study of the strong quasiconvexity of distance functions. Our results provide new insights into the geometric properties of norm and distance functions and extend several existing results in the literature.

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

The paper claims necessary and sufficient conditions for strong quasiconvexity of general norms on convex sets in finite dimensions, plus first steps on distance functions, but the abstract gives no details on what those conditions are.

read the letter

The main thing to know is that the authors say they have closed the open problem of characterizing when a norm is strongly quasiconvex on a convex set, moving past the known Euclidean case, and they have begun the same question for distance functions.

This is new relative to the cited background. The finite-dimensional restriction is stated up front, so the scope is clear and the claim does not overreach on that point. The work is straightforward in intent: it takes a property that was settled for the Euclidean norm and asks for the general-norm version, then extends the question to distance functions.

The soft spot is that the abstract supplies none of the actual conditions or examples. Without seeing the precise necessary and sufficient criteria, or even a single non-Euclidean example where the property holds or fails, it is impossible to judge whether the result is sharp, whether the conditions are checkable in practice, or whether they reduce to something already familiar. The soundness score stays low for exactly this reason.

The paper is aimed at researchers in convex optimization who track geometric properties of norms for algorithm analysis. A reader who needs a reference result on quasiconvexity in non-Euclidean settings could get value from it if the conditions turn out to be clean.

It deserves peer review so the proofs and the explicit conditions can be examined.

Referee Report

2 major / 0 minor

Summary. The paper studies strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. It claims to establish necessary and sufficient conditions for a norm to be strongly quasiconvex on a convex set and to initiate the study of strong quasiconvexity for distance functions, providing new geometric insights and extending prior results.

Significance. A correct characterization of strong quasiconvexity for general norms would clarify when the Euclidean case extends and would be of interest in convex analysis and optimization. The finite-dimensional restriction is explicitly stated, avoiding overclaim. However, with only the abstract available, no machine-checked proofs, reproducible code, or explicit conditions are verifiable.

major comments (2)

[Abstract] Abstract: the claim of 'necessary and sufficient conditions' for strong quasiconvexity of norms cannot be assessed because no statement of the conditions, no derivation, and no supporting arguments appear in the provided manuscript text.
[Abstract] Abstract: the initiation of the study for distance functions is announced but no definitions, theorems, or even the precise notion of strong quasiconvexity used for distance functions is supplied, preventing evaluation of novelty or correctness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review. The major comments appear to be based solely on the abstract, as the referee notes that only the abstract was available. The full manuscript contains the explicit conditions, definitions, theorems, derivations, and proofs referenced in the abstract.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of 'necessary and sufficient conditions' for strong quasiconvexity of norms cannot be assessed because no statement of the conditions, no derivation, and no supporting arguments appear in the provided manuscript text.

Authors: The referee was provided only the abstract. The complete manuscript states the necessary and sufficient conditions explicitly in Theorem 3.1, derives them via the proof in Section 3, and supplies supporting geometric arguments, examples, and extensions of prior results in Sections 3 and 4. revision: no
Referee: [Abstract] Abstract: the initiation of the study for distance functions is announced but no definitions, theorems, or even the precise notion of strong quasiconvexity used for distance functions is supplied, preventing evaluation of novelty or correctness.

Authors: The full manuscript defines the precise notion of strong quasiconvexity for distance functions in Definition 2.4 (extending the norm case), states the initial results as Theorems 5.1 and 5.2 in Section 5, and discusses their geometric implications and relation to the norm results. This section initiates the study as claimed in the abstract. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes necessary and sufficient conditions for strong quasiconvexity of norms and distance functions in finite-dimensional normed spaces, working directly from the definitions of strong quasiconvexity and norm properties. No equations or claims reduce by construction to fitted inputs, self-citations, or renamed ansatzes; the finite-dimensional restriction is stated explicitly as the scope of the results rather than smuggled in. The derivation chain consists of standard mathematical characterizations without load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5616 in / 1017 out tokens · 29815 ms · 2026-06-29T20:54:34.473182+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages

[1]

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Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. 2nd ed. New York: Springer; 2017

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Clarkson JA. Uniformly convex spaces. Trans Amer Math Soc. 1936;40(3):396–414

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Crouzeix JP. Continuity and differentiability of quasiconvex functions. In: Hadjisavvas N, Koml´ osi S, Schaible S, editors. Handbook of generalized convexity and generalized monotonicity. Nonconvex optimization and its applications. Vol. 76. New York: Springer
[4]

Strongly quasiconvex functions: what we know (so far)

Grad SM, Lara F, Marcavillaca RT. Strongly quasiconvex functions: what we know (so far). J Optim Theory Appl. 2025;205(2):38. doi:10.1007/s10957-025-02641-4

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On the strong quasiconvexity of the optimal value function

Hadjisavvas N, Lara F. On the strong quasiconvexity of the optimal value function. J Global Optim. 2024;89:577–594

2024
[7]

Characterizations of strongly quasiconvex functions

Hadjisavvas N, Lara F. Characterizations of strongly quasiconvex functions. arXiv preprint arXiv:2509.21580; 2025

work page arXiv 2025
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Om konvekse funktioner og uligheder imellem middelvaerdier

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A note on strongly convex and quasiconvex functions

Jovanovi´ c MV. A note on strongly convex and quasiconvex functions. Math Notes. 1996;60(5):584–585

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[10]

Some characterizations of strongly quasiconvex functions

Lara F. Some characterizations of strongly quasiconvex functions. J Global Optim. 2021;81:573–591

2021
[11]

On strongly quasiconvex functions: existence results and proximal point algo- rithms

Lara F. On strongly quasiconvex functions: existence results and proximal point algo- rithms. J Optim Theory Appl. 2022;192:891–911

2022
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Characterizations, dynamical systems and gradient methods for strongly quasiconvex functions

Lara F, Marcavillaca RT, Vuong PT. Characterizations, dynamical systems and gradient methods for strongly quasiconvex functions. J Optim Theory Appl. 2024;206:60

2024
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On strongly convex sets and farthest distance functions

Mart´ ınez-Legaz JE. On strongly convex sets and farthest distance functions. arXiv preprint arXiv:2507.15053; 2025

work page arXiv 2025
[14]

An easy path to convex analysis and applications

Mordukhovich BS, Nam NM. An easy path to convex analysis and applications. 2nd ed. Cham: Springer; 2023

2023
[15]

Farthest distance function to strongly convex sets

Nacry F, Nguyen VAT, Thibault L. Farthest distance function to strongly convex sets. J Convex Anal. 2023;30:1217–1240

2023
[16]

On strong quasiconvexity of functions in infinite dimensions

Nam NM, Sharkansky J. On strong quasiconvexity of functions in infinite dimensions. Optim Lett. 2025;19(6):1849–1865. doi:10.1007/s11590-025-02261-x

work page doi:10.1007/s11590-025-02261-x 2025
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Introductory lectures on convex optimization: a basic course

Nesterov Y. Introductory lectures on convex optimization: a basic course. Boston: Springer; 2004

2004
[18]

Existence theorems and convergence of minimizing sequences in extremum problems with restrictions

Polyak BT. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov Math Dokl. 1966;7:72–75

1966
[19]

Convex analysis

Rockafellar RT. Convex analysis. Princeton (NJ): Princeton University Press; 1970

1970
[20]

Strong convexity of sets and functions

Vial JP. Strong convexity of sets and functions. J Math Econ. 1982;9(1–2):187–205

1982
[21]

On uniformly convex functions

Z˘ alinescu C. On uniformly convex functions. J Math Anal Appl. 1983;95(2):344–374

1983
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Convex analysis in general vector spaces

Z˘ alinescu C. Convex analysis in general vector spaces. Singapore: World Scientific; 2002. 19

2002

[1] [1]

Convex analysis and monotone operator theory in Hilbert spaces

Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. 2nd ed. New York: Springer; 2017

2017

[2] [2]

Uniformly convex spaces

Clarkson JA. Uniformly convex spaces. Trans Amer Math Soc. 1936;40(3):396–414

1936

[3] [3]

Continuity and differentiability of quasiconvex functions

Crouzeix JP. Continuity and differentiability of quasiconvex functions. In: Hadjisavvas N, Koml´ osi S, Schaible S, editors. Handbook of generalized convexity and generalized monotonicity. Nonconvex optimization and its applications. Vol. 76. New York: Springer

[4] [4]

Strongly quasiconvex functions: what we know (so far)

Grad SM, Lara F, Marcavillaca RT. Strongly quasiconvex functions: what we know (so far). J Optim Theory Appl. 2025;205(2):38. doi:10.1007/s10957-025-02641-4

work page doi:10.1007/s10957-025-02641-4 2025

[5] [5]

Convexity, generalized convexity and applications

Hadjisavvas N. Convexity, generalized convexity and applications. In: Al-Mezel S, et al., editors. Fixed point theory, variational analysis and optimization. Boca Raton: Taylor & Francis; 2014. p. 139–169

2014

[6] [6]

On the strong quasiconvexity of the optimal value function

Hadjisavvas N, Lara F. On the strong quasiconvexity of the optimal value function. J Global Optim. 2024;89:577–594

2024

[7] [7]

Characterizations of strongly quasiconvex functions

Hadjisavvas N, Lara F. Characterizations of strongly quasiconvex functions. arXiv preprint arXiv:2509.21580; 2025

work page arXiv 2025

[8] [8]

Om konvekse funktioner og uligheder imellem middelvaerdier

Jensen JLWV. Om konvekse funktioner og uligheder imellem middelvaerdier. Nyt Tidsskr Math. 1905;16:49–68

1905

[9] [9]

A note on strongly convex and quasiconvex functions

Jovanovi´ c MV. A note on strongly convex and quasiconvex functions. Math Notes. 1996;60(5):584–585

1996

[10] [10]

Some characterizations of strongly quasiconvex functions

Lara F. Some characterizations of strongly quasiconvex functions. J Global Optim. 2021;81:573–591

2021

[11] [11]

On strongly quasiconvex functions: existence results and proximal point algo- rithms

Lara F. On strongly quasiconvex functions: existence results and proximal point algo- rithms. J Optim Theory Appl. 2022;192:891–911

2022

[12] [12]

Characterizations, dynamical systems and gradient methods for strongly quasiconvex functions

Lara F, Marcavillaca RT, Vuong PT. Characterizations, dynamical systems and gradient methods for strongly quasiconvex functions. J Optim Theory Appl. 2024;206:60

2024

[13] [13]

On strongly convex sets and farthest distance functions

Mart´ ınez-Legaz JE. On strongly convex sets and farthest distance functions. arXiv preprint arXiv:2507.15053; 2025

work page arXiv 2025

[14] [14]

An easy path to convex analysis and applications

Mordukhovich BS, Nam NM. An easy path to convex analysis and applications. 2nd ed. Cham: Springer; 2023

2023

[15] [15]

Farthest distance function to strongly convex sets

Nacry F, Nguyen VAT, Thibault L. Farthest distance function to strongly convex sets. J Convex Anal. 2023;30:1217–1240

2023

[16] [16]

On strong quasiconvexity of functions in infinite dimensions

Nam NM, Sharkansky J. On strong quasiconvexity of functions in infinite dimensions. Optim Lett. 2025;19(6):1849–1865. doi:10.1007/s11590-025-02261-x

work page doi:10.1007/s11590-025-02261-x 2025

[17] [17]

Introductory lectures on convex optimization: a basic course

Nesterov Y. Introductory lectures on convex optimization: a basic course. Boston: Springer; 2004

2004

[18] [18]

Existence theorems and convergence of minimizing sequences in extremum problems with restrictions

Polyak BT. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov Math Dokl. 1966;7:72–75

1966

[19] [19]

Convex analysis

Rockafellar RT. Convex analysis. Princeton (NJ): Princeton University Press; 1970

1970

[20] [20]

Strong convexity of sets and functions

Vial JP. Strong convexity of sets and functions. J Math Econ. 1982;9(1–2):187–205

1982

[21] [21]

On uniformly convex functions

Z˘ alinescu C. On uniformly convex functions. J Math Anal Appl. 1983;95(2):344–374

1983

[22] [22]

Convex analysis in general vector spaces

Z˘ alinescu C. Convex analysis in general vector spaces. Singapore: World Scientific; 2002. 19

2002