pith. sign in

arxiv: 2606.07887 · v1 · pith:OZNWRC2Inew · submitted 2026-06-05 · 🧮 math.MG

Equality cases for the L_p-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in mathbb{R}^n

Matthieu Fradelizi , Auttawich Manui , Mark Meyer , Cheikh Saliou Ndiaye This is my paper

Pith reviewed 2026-06-27 19:53 UTC · model grok-4.3

classification 🧮 math.MG
keywords L_p-Rogers-Shephard inequalityequality casesconvex bodiessimplexlocally anti-blocking bodiesvolume inequalitiesdifference body
0
0 comments X

The pith

Equality in the L_p-Rogers-Shephard inequality holds exactly when the convex body is a simplex with one vertex at the origin.

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

The paper determines the equality cases for the L_p-Rogers-Shephard inequality in two settings: the plane and locally anti-blocking bodies in any dimension. For p greater than 1 the bound is attained if and only if the convex body is a simplex with one vertex at the origin. A reader cares because equality cases identify the unique shapes that saturate a volume comparison between a body and its L_p difference body. The result completes the description of these inequalities by specifying when the bound is sharp.

Core claim

For p>1, equality holds in the L_p-Rogers-Shephard inequality if and only if the convex body is a simplex with one vertex at the origin. This applies both to the planar case and to locally anti-blocking convex bodies in R^n.

What carries the argument

The simplex with one vertex at the origin, identified as the sole body attaining equality through direct analysis of the volume inequality.

If this is right

  • Equality is attained if and only if the body is a simplex with a vertex at the origin.
  • The characterization is the same in both the planar setting and the locally anti-blocking setting.
  • For p>1 the inequality is strict for every other convex body.
  • The extremal bodies are independent of the specific value of p as long as p exceeds 1.

Where Pith is reading between the lines

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

  • The same equality bodies may appear in other L_p-type volume inequalities for convex bodies.
  • Direct volume computation on a simplex with origin vertex in low dimensions would confirm attainment of equality.
  • The result suggests testing whether analogous equality statements hold when the anti-blocking assumption is dropped.

Load-bearing premise

The L_p-Rogers-Shephard inequalities hold with the bounds given in prior statements.

What would settle it

A convex body that is not a simplex with one vertex at the origin yet attains equality in the inequality for some p>1.

read the original abstract

The classical Rogers--Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes the equality cases for two previously established L_p-Rogers-Shephard inequalities: the planar version due to Bianchini-Colesanti and the version for locally anti-blocking convex bodies in R^n due to Zvavitch et al. The central claim is that, for p>1, equality holds if and only if the convex body is a simplex with one vertex at the origin.

Significance. If the characterization holds, it completes the analysis of these inequalities by identifying the extremal bodies, a standard and valuable contribution in convex geometry. The result is sharp, recovers the classical Rogers-Shephard equality cases in the limit p o1, and is stated in a parameter-free manner that directly identifies the equality bodies without additional constants.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of Theorem 1.1 could explicitly recall the precise form of the L_p-Rogers-Shephard inequality being referred to, for the reader's convenience.
  2. [§3] §3 (Proof of the planar case): the transition from the inequality established in Bianchini-Colesanti to the equality analysis would benefit from a short sentence clarifying which steps rely on strict convexity of the L_p sum for p>1.
  3. [Preliminaries] Notation: the symbol for the L_p sum is used consistently but could be defined once more explicitly in the preliminaries section to avoid any ambiguity with the classical Minkowski sum.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

Minor self-citation to prior inequality results; equality characterization independent

full rationale

The paper assumes the L_p-Rogers-Shephard inequalities as previously proved by Bianchini-Colesanti (plane case) and Zvavitch et al. (locally anti-blocking bodies, overlapping with current authors on the second and fourth positions) and then derives the equality cases for p>1, identifying simplices with a vertex at the origin. This self-citation supports the base inequalities but is not load-bearing for the new equality characterization, which adds independent content via analysis of those statements. No self-definitional reductions, fitted inputs renamed as predictions, uniqueness theorems imported from the same authors, or other enumerated circular patterns appear in the derivation chain. The central claim remains externally grounded and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal; no free parameters, invented entities, or non-standard axioms are visible.

axioms (1)
  • domain assumption The L_p-Rogers-Shephard inequalities hold as stated in the cited prior works for the indicated classes of bodies.
    The equality analysis is built directly on those inequalities.

pith-pipeline@v0.9.1-grok · 5629 in / 1117 out tokens · 19174 ms · 2026-06-27T19:53:40.989514+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    Alonso-Guti´errez,A reverse Rogers-Shephard inequality for log-concave functions, J

    D. Alonso-Guti´errez,A reverse Rogers-Shephard inequality for log-concave functions, J. Geom. Anal., 29 (2019), pp. 299–315

    2019

  2. [2]

    Alonso-Guti´errez, S

    D. Alonso-Guti´errez, S. Artstein-Avidan, B. Gonz´alez Merino, C. H. Jim ´enez, and R. Villa, Rogers-Shephard and local Loomis-Whitney type inequalities, Math. Ann., 374 (2019), pp. 1719–1771

    2019

  3. [3]

    Alonso-Guti´errez, B

    D. Alonso-Guti´errez, B. Gonz ´alez Merino, C. H. Jim ´enez, and R. Villa,Rogers-Shephard inequality for log-concave functions, J. Funct. Anal., 271 (2016), pp. 3269–3299

    2016

  4. [4]

    Alonso-Guti´errez, M

    D. Alonso-Guti´errez, M. A. Hern´andez Cifre, M. Roysdon, J. Yepes Nicol´as, and A. Zvavitch, On Rogers-Shephard type inequalities for general measures, Int. Math. Res. Not. IMRN, (2021), pp. 7224– 7261

    2021

  5. [5]

    Artstein-Avidan, S

    S. Artstein-Avidan, S. Sadovsky, and R. Sanyal,Geometric inequalities for anti-blocking bodies, Communications in Contemporary Mathematics, 25 (2023), p. 2150113

    2023

  6. [6]

    Bianchini and A

    C. Bianchini and A. Colesanti,A sharp Rogers and Shephard inequality for the p-difference body of planar convex bodies, Proc. Amer. Math. Soc., 136 (2008), pp. 2575–2582. [7]S. Campi, A. Colesanti, and P. Gronchi,A note on Sylvester’s problem for random polytopes in a convex body, Rend. Istit. Mat. Univ. Trieste, 31 (1999), pp. 79–94

    2008

  7. [7]

    Campi and P

    S. Campi and P. Gronchi,The Lp-Busemann-Petty centroid inequality, Adv. Math., 167 (2002), pp. 128–141

    2002

  8. [8]

    Campi and P

    S. Campi and P. Gronchi,Volume inequalities for Lp-zonotopes, Mathematika, 53 (2006), pp. 71–80

    2006

  9. [9]

    Publ., Hackensack, NJ, 2006, pp

    ,Volume inequalities for sets associated with convex bodies, in Integral geometry and convexity, World Sci. Publ., Hackensack, NJ, 2006, pp. 1–15

    2006

  10. [10]

    Chappell, T

    T. Chappell, T. Friedl, and R. Sanyal,Two double poset polytopes, SIAM J. Discrete Math., 31 (2017), pp. 2378–2413

    2017

  11. [11]

    Colesanti,Functional inequalities related to the Rogers-Shephard inequality, Mathematika, 53 (2006), pp

    A. Colesanti,Functional inequalities related to the Rogers-Shephard inequality, Mathematika, 53 (2006), pp. 81–101. [13]W. J. Firey,p-means of convex bodies, Math. Scand., 10 (1962), pp. 17–24

    2006

  12. [12]

    Fradelizi, A

    M. Fradelizi, A. Manui, M. Meyer, and C. S. Ndiaye,On Lp-Rogers-Shephard type inequalities forL p-zonoids and symmetric bodies. In preparation

  13. [13]

    D. R. Fulkerson,Blocking and anti-blocking pairs of polyhedra, Math. Programming, 1 (1971), pp. 168– 194

    1971

  14. [14]

    P. C. Hammer,Convex bodies associated with a convex body, Proc. Amer. Math. Soc., 2 (1951), pp. 781–793

    1951

  15. [15]

    ,Convex curves of constant Minkowski breadth, in Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, RI, 1963, pp. 291–304

    1963

  16. [16]

    Kotrbat´y,On a generalization of Godbersen’s conjecture, International Mathematics Research Notices, 2025 (2025), p

    J. Kotrbat´y,On a generalization of Godbersen’s conjecture, International Mathematics Research Notices, 2025 (2025), p. rnaf341

    2025

  17. [17]

    Lutwak,The Brunn-Minkowski-Firey theory

    E. Lutwak,The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), pp. 131–150

    1993

  18. [18]

    ,The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math., 118 (1996), pp. 244–294

    1996

  19. [19]

    Lutwak, D

    E. Lutwak, D. Yang, and G. Zhang,The Brunn-Minkowski-Firey inequality for nonconvex sets, Adv. in Appl. Math., 48 (2012), pp. 407–413

    2012

  20. [20]

    Madiman, A

    M. Madiman, A. Manui, B. Zawalski, and A. Zvavitch,Integral inequalities for α-convolutions of α-concave functions. In preparation

  21. [21]

    Manui, C

    A. Manui, C. S. Ndiaye, and A. Zvavitch,On the volume of sums of anti-blocking convex bodies, Communications in Contemporary Mathematics, 22 (2025), p. 2550095

    2025

  22. [22]

    Meyer,Convex bodies with minimal volume product inR 2, Monatsh

    M. Meyer,Convex bodies with minimal volume product inR 2, Monatsh. Math., 112 (1991), pp. 297–301

    1991

  23. [23]

    C. A. Rogers and G. C. Shephard,The difference body of a convex body, Arch. Math. (Basel), 8 (1957), pp. 220–233

    1957

  24. [24]

    London Math

    ,Convex bodies associated with a given convex body, J. London Math. Soc., 33 (1958), pp. 270–281

    1958

  25. [25]

    ,Some extremal problems for convex bodies, Mathematika, 5 (1958), pp. 93–102

    1958

  26. [26]

    Sadovsky,Godbersen’s conjecture for locally anti-blocking bodies, Advances in Geometry, 25 (2025), pp

    S. Sadovsky,Godbersen’s conjecture for locally anti-blocking bodies, Advances in Geometry, 25 (2025), pp. 307–315

    2025

  27. [27]

    Schneider,Convex bodies: the Brunn-Minkowski theory, vol

    R. Schneider,Convex bodies: the Brunn-Minkowski theory, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993. [30]V. Soltan,Affine diameters of convex-bodies—a survey, Expo. Math., 23 (2005), pp. 47–63. EQUALITY CASES FOR THEL p-ROGERS–SHEPHARD INEQUALITY 17 Matthieu Fradelizi LAMA, Univ Gustave Eiffel, ...

    1993