Home Mathematics A direct proof of the characterization of the convexity of the discrete Choquet integral
Article
Licensed
Unlicensed Requires Authentication

A direct proof of the characterization of the convexity of the discrete Choquet integral

  • José Carlos R. Alcantud ORCID logo EMAIL logo
Published/Copyright: December 12, 2025
Become an author with De Gruyter Brill

Abstract

This article presents the first self-contained and direct proof of a widely recognized result that the discrete Choquet integral, when defined from a discrete fuzzy measure (or capacity), is convex if and only if the discrete fuzzy measure itself is submodular. In contrast to existing proofs, our argument is constructed directly from the fuzzy measure defined on a finite set, employing only standard techniques from the theory of capacities and Choquet integration.


Financial support of the Department of Education of the Junta de Castilla y León and FEDER Funds (Reference: CLU-2O25-2-03) is gratefully acknowledged.


  1. (Communicated by Anatolij Dvurečenskij)

References

[1] Alcantud, J. C. R.: A characterization of delay averse Choquet integrals for intertemporal analysis, Theor. Decis. (2025), forthcoming.10.1007/s11238-025-10047-xSearch in Google Scholar

[2] Alcantud, J. C. R.: New classes of differentially private Choquet integrals for privacy preservation, submitted.Search in Google Scholar

[3] Alcantud, J. C. R.—De Andrés, R.: The problem of collective identity in a fuzzy environment, Fuzzy Sets and Systems 315 (2017), 57–75.10.1016/j.fss.2016.02.011Search in Google Scholar

[4] Alfonsi, A.: A simple proof for the convexity of the Choquet integral, Stat. Probab. Lett. 104 (2015), 22–25.10.1016/j.spl.2015.04.022Search in Google Scholar

[5] Anger, B.: Representation of capacities, Math. Ann. 229 (1977), 245–258.10.1007/BF01391470Search in Google Scholar

[6] Beliakov, G.—James, S.: Choquet integral-based measures of economic welfare and species diversity, Int. J. Intell. Syst. 37 (2022), 2849–2867.10.1002/int.22609Search in Google Scholar

[7] Buja, A.: Simultaneously least favorable experiments, Z. Wahrsch. Verw. Gebiete 65 (1984), 367–384.10.1007/BF00533742Search in Google Scholar

[8] Bastianello, L.—Faro, J. E.: Choquet expected discounted utility, J. Econom. Theory 75 (2023), 1071–1098.10.1007/s00199-022-01438-0Search in Google Scholar

[9] Bustince, H.—Mesiar, R.—Fernández, J.—Galar, M.—Paternain, D.—Altalhi, A.—Dimuro, G. P.—Bedregal, B.—Takáč, Z.: d-Choquet integrals: Choquet integrals based on dissimilarities, Fuzzy Sets and Systems 414 (2021), 1–27.10.1016/j.fss.2020.03.019Search in Google Scholar

[10] De Campos Ibáñez, L. M.—Bolaños Carmona, M. J.: Representation of fuzzy measures through probabilities, Fuzzy Sets and Systems 31 (1989), 23–36.10.1016/0165-0114(89)90064-XSearch in Google Scholar

[11] Candeloro, D.—Mesiar, R.—Sambucini, A. R.: A special class of fuzzy measures: Choquet integral and applications, Fuzzy Sets and Systems 355 (2019), 83–99.10.1016/j.fss.2018.04.008Search in Google Scholar

[12] Chakrabarty, D.—Huang, Z.: Testing coverage functions. In: Automata, Languages, and Programming (A. Czumajet al., eds.), Springer Berlin Heidelberg, 2012, pp. 170–181.10.1007/978-3-642-31594-7_15Search in Google Scholar

[13] Choquet, G.: Theory of capacities, Ann. Inst. Fourier 5 (1954), 131–295.10.5802/aif.53Search in Google Scholar

[14] Denneberg, D.: Non-Additive Measure and Integral, Springer Science & Business Media, 2013.Search in Google Scholar

[15] Iwata, S.—Fleischer, L.—Fujishige, S.: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions, J. ACM 48 (2001), 761–777.10.1145/502090.502096Search in Google Scholar

[16] Frank, A.: Submodular functions in graph theory. Discrete Math. 111 (1993), 231–243.10.1016/0012-365X(93)90158-PSearch in Google Scholar

[17] Grabisch, M.—Marichal, J.-L.—Mesiar, R.—Pap, E.: Aggregation Functions, Cambridge University Press, 2009.10.1017/CBO9781139644150Search in Google Scholar

[18] Grabisch, M.—Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid, Ann. Oper. Res. 175 (2010), 247–290.10.1007/s10479-009-0655-8Search in Google Scholar

[19] Huber, P. J.: Robust Statistics, John Wiley & Sons, 1981.10.1002/0471725250Search in Google Scholar

[20] Iyer, R.—Khargoankar, N.—Bilmes, J.—Asanani, H.: Submodular combinatorial information measures with applications in machine learning. In: Proceedings of the 32nd International Conference on Algorithmic Learning Theory (V. Feldman, K. Ligett, S. Sabato, eds.), 2021, pp. 722–754.Search in Google Scholar

[21] Lovász, L.: Submodular functions and convexity. In: Mathematical Programming The State of the Art (A. Bachem, B. Korte, M. Grötschel, eds.), 1983, p. 235–257.10.1007/978-3-642-68874-4_10Search in Google Scholar

[22] Ok, E. A.—Zhou, L..: The Choquet bargaining solutions, Games Econ. Behav. 33 (2000), 249–264.10.1006/game.1999.0778Search in Google Scholar

[23] Sambucini, A. R.: The Choquet integral with respect to fuzzy measures and applications, Math. Slovaca 67 (2017), 1427–1450.10.1515/ms-2017-0049Search in Google Scholar

[24] Takáč, Z.—Uriz, M.—Galar, M.—Paternain, D.—Bustince, H.: Discrete IV dG-Choquet integrals with respect to admissible orders, Fuzzy Sets and Systems 441 (2022), 169–195.10.1016/j.fss.2021.09.013Search in Google Scholar

[25] Torra, V.: Differentially private Choquet integral: extending mean, median, and order statistics, Int. J. Inf. Secur. 24 (2025), 68.10.1007/s10207-025-00984-7Search in Google Scholar

[26] Torra, V.—Narukawa, Y.: The h-index and the number of citations: Two fuzzy integrals, IEEE Trans. Fuzzy Syst. 16 (2008), 795–797.10.1109/TFUZZ.2007.896327Search in Google Scholar

Received: 2025-07-05
Accepted: 2025-07-31
Published Online: 2025-12-12
Published in Print: 2025-12-17

© 2025 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. A new categorical equivalence for stone algebras
  2. On special classes of prime filters in BL-algebras
  3. A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
  4. New Young-type integral inequalities using composition schemes
  5. The structure of pseudo-n-uninorms with continuous underlying functions
  6. Jensen-type inequalities for a second-order differential inequality condition
  7. A direct proof of the characterization of the convexity of the discrete Choquet integral
  8. Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
  9. Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
  10. Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
  11. Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
  12. Weighted B-summability and positive linear operators
  13. Some properties and applications of convolution algebras
  14. On measures of σ-noncompactess in F-spaces
  15. On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
  16. Intermediately trimmed sums of oppenheim expansions: A strong law
  17. Novel weighted distribution: Properties, applications and web-tool
  18. On the q-Gamma distribution: Properties and inference
  19. Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
  20. Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0099/html
Scroll to top button