Extremal functions for real convex bodies: simplices, strips, and ellipses

Sione Ma`u

REVIEW 2 major objections 3 minor 17 references

Extremal functions for real convex bodies: simplices, strips, and ellipses

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every point outside a real convex body lies on an extremal complex ellipse.

desk verdict Solid, mostly elementary paper with a genuinely new polytope decomposition and a minor but easily patched gap in the final approximation argument. read the letter →

arxiv 1908.07118 v3 pith:TAM72TPW submitted 2019-08-20 math.CV

Sione Ma`u This is my paper · ORCID

classification math.CV MSC 32U0532U1552A20

keywords Siciak-ZaharjutaextremalfunctionpluripotentialtheoryrealconvexbodypolytopeellipseRobinexponentialmapbarycentriccoordinates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes an explicit decomposition for the Siciak-Zaharjuta extremal function of a real convex polytope: it is the maximum of the extremal functions of a finite list of supporting simplices and strips. This reduces polytope computations to barycentric-coordinate formulas on simplices together with a dimension-reduction formula for strips. Using this polytope result, the author gives a new proof that for every real convex body and every exterior point there is a complex ellipse through that point on which the extremal function equals $\log|\zeta|$, with the real ellipse inscribed in the body. The significance is a self-contained route to the ellipse foliation that avoids the earlier geodesic-based construction.

What carries the argument

The load-bearing identity is $V_K(z)=\max\{V_S(z):S\in\mathcal S(K)\}$ (Theorem 5.4), where $\mathcal S(K)$ is the finite collection of supporting simplices and strips whose intersection is $K$. For a simplex, the extremal function is $V_S(z)=\log h(|\lambda_0(z)|+\cdots+|\lambda_d(z)|)$, with $\lambda_j$ the barycentric coordinates and $h(\eta)=\eta+\sqrt{\eta^2-1}$ the inverse Joukowski map. Strips are handled by the reduction $V_{K\times \mathbb R^{d-j}}(z',z'')=V_K(z')$, proved by a ball-limit argument. The ellipse side is carried by the Robin exponential map $R_K(c\zeta)=a(c)+c\zeta+c/\zeta$, where $a(c)$ is the center of the unique inscribed ellipse of shape $c$; for the ball this is a generalized Joukowski map, and for the simplex it is obtained from the ball case by the square map.

What would settle it

For a concrete test, take the unit square in $\mathbb R^2$, choose an exterior point such as $z=(2+i,3)$, solve the equations $z=a+c\zeta+c/\zeta$ numerically for $a\in\mathbb R^2$, $c\in\mathbb C^2$, and $|\zeta|>1$, and compare $\log|\zeta|$ with a high-degree polynomial-envelope approximation of $V_K(a+c\zeta+c/\zeta)$. A nonzero difference for every such solution would falsify the theorem.

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Extended reading notes

Core claim

The central claim is that the extremal function $V_K$ of a compact convex polytope is fully determined by supporting simplices and strips: $V_K(z)=\max\{V_S(z): S\in\mathcal S(K)\}$, where the sets in $\mathcal S(K)$ are finitely many simplices and strips whose common intersection is $K$. For a simplex, $V_S$ has the closed form $\log h(|\lambda_0(z)|+\cdots+|\lambda_d(z)|)$ in terms of barycentric coordinates and the inverse Joukowski function; for a strip, $V_S$ reduces to the extremal function of a lower-dimensional cross-section. From this the paper constructs, first for the ball, then for a simplex, then for a polytope, a foliation of $\mathbb C^d\setminus K$ by complexified ellipses on which $V_K=\log|\zeta|$, and obtains the general convex-body case by decreasing polytopal approximation.

Load-bearing premise

The proof assumes that the extremal functions of the approximating polytopes converge, point by point, to the extremal function of the convex body they shrink to, and that the corresponding inscribed ellipses have a locally uniform limit; the entire convex-body theorem rests on that convergence.

Editorial extensions

If this is right

For every compact convex polytope, the extremal function is explicitly computable as a maximum of barycentric-coordinate formulas over a finite list of supporting simplices and strips.
For polytopes satisfying the face-normal independence condition, the Robin exponential map is a homeomorphism from $\mathbb C^d\setminus K_\rho$ to $\mathbb C^d\setminus K$ whose leaves are complexified ellipses on which $V_K=\log|\zeta|$.
For any compact convex body, each exterior point lies on a complex ellipse whose real part is an inscribed real ellipse and on which the extremal function is exactly $\log|\zeta|$.
The paper notes that known results on the regularity of $V_K$ and on the complex equilibrium measure $(dd^c V_K)^d$ follow from this self-contained theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the construction is algorithmic in nature: given the half-space description of a polytope, the list of supporting simplices and strips is produced by rank checks on normal vectors, so $V_K$ and the equilibrium measure could be evaluated symbolically for any polytope.
Editorial inference: the strip factorisation suggests that when face normals are linearly dependent, the computation genuinely lowers dimension; the polytope problem projects to a lower-dimensional simplex problem, which may explain why strips require no separate analytic machinery beyond the simplex formula.
Editorial inference: the identity $\log|\zeta|=V_K(a+c\zeta+c/\zeta)$ is concrete enough to be a numerical check of the theory for arbitrary bodies: choose a body, choose an exterior point, solve for the ellipse data, and compare the two sides by polynomial approximation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Ma'u's paper is a genuinely useful piece of elementary pluripotential geometry. The main new result is Theorem 5.4: for a compact convex polytope, the Siciak-Zaharjuta extremal function is the maximum of the extremal functions of finitely many supporting simplices and strips. That decomposition is real and new as far as I can tell, and the convex-geometric construction behind it is worked out in satisfying detail. The second half builds ellipses on which V_K is harmonic, first for balls and simplices, then for polytopes via the Robin exponential map, and finally for general convex bodies by approximation. The upshot is a new proof of the Burns-Levenberg-Ma'u existence theorem that avoids Lempert theory entirely. That is a legitimate contribution: the paper makes the existence of extremal ellipses more accessible and gives an explicit computational recipe for polytopes.

The soft spots are concentrated in the last step. In Theorem 13.1, after taking the normal-families limit f_j→f and ζ_j→ζ_z, the text claims (i) and (ii) hold, but the displayed computation only establishes V_K(f(ζ_z)) = log|ζ_z| at that single parameter value. Condition (ii) requires equality for every |ζ|>1. The missing argument is standard: monotone convergence of continuous extremal functions gives local uniform convergence (Dini), and then uniform convergence on compact annuli transfers the equalities from the approximating ellipses to the limit. The paper even notes uniform convergence on compact annular regions, but does not connect the dots. A referee should ask for that one paragraph. The other asserted ingredient, that the approximating polytopes can be chosen with the generic linear-independence condition and with a_j, c_j bounded, is also true but under-documented; it needs a sentence or two.

None of this shakes the central argument. Section 5.4 is the real result, and it is proved cleanly. The reliance on earlier papers, including the author's own [4] and [6], is for standard formulas (barycentric coordinates, Robin exponential terminology) rather than for the main novelty, so I do not see a circularity problem. The paper is honest about what it reproves and what is new.

This is a paper I would send to a competent referee. It deserves a careful reading and likely a minor revision. I would cite it if I worked on extremal functions for real convex bodies, and I would bring it to our reading group.

Referee Report

2 major / 3 minor

Summary. The paper develops an explicit method for computing the Siciak–Zaharjuta extremal function of a compact convex polytope in R^d in terms of a finite family of supporting simplices and strips, and uses this to reprove the existence of extremal ellipses for general real convex bodies. The main theorems are Theorem 5.4, asserting V_K(z)=max{V_S(z): S∈S(K)} for polytopes, and Theorem 13.1, asserting that for every z outside a compact convex body K there is a complexified ellipse through z on which V_K equals log|ζ| and whose real trace is inscribed in K. The proof route goes through geometric lemmas on simplices and strips, explicit extremal functions for balls and simplices, Hooke and Newton ellipses, the Robin exponential map for polytopes, and a final approximation argument from polytopes to general convex bodies.

Significance. If the main results are correct, the paper provides a self-contained and largely elementary route to structural facts about extremal functions for real convex bodies that were previously obtained using Lempert theory. The explicit barycentric-coordinate formulas and the constructive polytope algorithm in Section 7 are valuable, as is the new derivation of Lundin and Baran formulas through the square map. The Robin exponential map for polytopes and the approximation scheme for general convex bodies are potentially useful tools. However, the proof of the central polytope formula contains a false convex-geometric step, and the approximation step in Theorem 13.1 has a gap; both need to be addressed before the main claims are established.

major comments (2)

[Section 5, proof of Theorem 5.4] The step 'The set Π is a product of convex sets, hence convex, and therefore Π⊂π_Δ^{-1}(K_Δ)' is false: orthogonal projection of a convex set onto a subspace need not be contained in the set. The paper's own Section 7 example gives a concrete counterexample. Let S_3={x_1≥0, x_2≥0, 3−x_1−3x_2≥0} and S_4={x_1≥0, x_2≥0, 3−3x_1−x_2≥0}; these are the two supporting triangles in S(K) for the quadrilateral of Section 7. The points x=(2,0.2)∈S_3 and y=(0.2,2)∈S_4 satisfy π_Δ(x,y)=((1.1,1.1),(1.1,1.1)), and (1.1,1.1) lies in neither S_3 nor S_4, so π_Δ(x,y)∉K_Δ even though (x,y)∈Π. Thus the claimed inclusion Π⊂π_Δ^{-1}(K_Δ) fails in the exact class of examples considered. Since the reverse inequality V_K≤max_j V_{S_j} is the core of Theorem 5.4 and is used later in Lemma 8.2 and in the Robin exponential map construction, this is a load-bearing gap in the proof.
[Section 13, proof of Theorem 13.1] The normal-families argument proves V_K(f(ζ_z))=log|ζ_z| only at the single parameter value ζ_z, whereas condition (ii) requires equality for every ζ with |ζ|>1. The sentence 'Hence (i) and (ii) hold' is not justified by pointwise convergence alone. The gap is repairable: since the sets K_j and K are regular by Lemma 5.2, the functions V_{K_j} and V_K are continuous, and Proposition 5.1(3) gives monotone pointwise convergence V_{K_j}↗V_K; Dini's theorem on compact sets then gives locally uniform convergence, and combined with f_j→f locally uniformly this yields V_K(f(ζ))=log|ζ| for all |ζ|>1. As written, however, the proof is incomplete at a point that is essential for all applications of Theorem 13.1.

minor comments (3)

[Section 5, Proposition 5.1(2)] The statement says 'polynomial mapping of degree d' and gives the formula V_{P^{-1}(K)}(z)=1/d V_K(P(z)), but the correct factor is the reciprocal of the degree of the polynomial mapping, not the dimension d. As stated, the proposition is false for a linear map when d>1. The later use in Corollary 6.4 correctly uses the factor 1/2 for the degree-2 square map, so this is likely a typographical error, but it should be corrected.
[Section 7] The labels l_3 and l_4 for the two non-axis supporting lines are interchanged between Remark 3.8 and Section 7: Remark 3.8 has l_3(x)=3−3x_1−x_2 and l_4(x)=3−x_1−3x_2, while Section 7 defines l_3(z)=3−z_1−3z_2 and l_4(z)=3−3z_1−z_2. Please align the notation to avoid confusion in checking the example.
[Section 13] The existence of a decreasing sequence of polytopes K_j satisfying the linear independence condition of Theorem 3.9 and converging to K is asserted without proof. This is a standard genericity/refinement argument, but since the convergence step of Theorem 13.1 depends on it, a brief justification would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained given standard pluripotential theory, and the same-author citations are parameter-free ingredients rather than repackaged conclusions.

full rationale

The central derivation chain is not circular. Theorem 5.4 proves VK = max VS for supporting simplices and strips using Lemma 5.3 and Proposition 5.1; the reverse inequality is obtained by embedding K as a diagonal slice of a product of strips and applying the product formula, not by assuming the conclusion. The simplex extremal formula (Theorem 6.2) is cited from [4], which has overlapping authorship, but it is a parameter-free external theorem with its own stated assumptions and does not include the target ellipse-existence result, so under the stated rules it counts as independent support rather than circularity. Likewise, the use of [3] for continuity of WK across the hyperplane at infinity is an ingredient, not the result being derived. Theorem 13.1 does contain a genuine proof gap: the line 'By the pointwise convergence VKj↗VK (Proposition 5.1(3)) we have VK(f(ζz)) = ... Hence (i) and (ii) hold' establishes equality only at the single parameter value ζz, while condition (ii) requires equality for all |ζ| > 1; a maximum-principle or locally-uniform convergence argument is needed to propagate the equality. This is an incompleteness in the proof, not a circularity, and no equation in the paper reduces a claimed prediction to an input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new postulated entities. Its foundational inputs are standard results from pluripotential theory, cited to Klimek's textbook: the Siciak-Zaharjuta envelope theorem (5.1), the product formula and polynomial mapping formula (Proposition 5.1), and the continuity of extremal functions for regular sets. The extremal function formulas for simplices and balls are imported from prior literature ([4] for the barycentric formula, Lundin and Baran for the standard simplex), which serve as axioms in this paper's economy rather than being derived. These are documented in the axiom_audit field.

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Cite this review

Pith. "Pith review of Extremal functions for real convex bodies: simplices, strips, and ellipses." pith.science (2026). https://pith.science/paper/TAM72TPW

@misc{pith2026190807118,
  author       = {Pith},
  title        = {Pith review of: Extremal functions for real convex bodies: simplices, strips, and ellipses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAM72TPW}},
  note         = {Machine review of arXiv:1908.07118}
}

read the original abstract

We present an explicit method to compute the (Siciak-Zaharjuta) extremal function of a real convex polytope in terms of supporting simplices and strips. We use this to give a new proof of the existence of extremal ellipses associated to the extremal function of a real convex body.

Figures

Figures reproduced from arXiv: 1908.07118 by the authors.

**Figure 2.** Theorem 3.9 in R 3 , with K = S1 ∩ S2 and S1 = co{p0, p1, p2, p3}. Now p0 is a vertex of at least d faces, F1, . . . , Fd say. Let H1, . . . , Hd and l1, . . . , ld be the corresponding supporting hyperplanes and affine maps as above. Let ζ ∈ F0. By convexity, the line segment joining p0 and ζ lies in K. Let x0 be the midpoint of this segment. Then by (3.6), l0(p0) > 0 = l0(ζ), so l0 decreases linearly along the seg… view at source ↗

**Figure 3.** Examples of strips in R 3 . Proof. Let b be a vector. The simplex S has a supporting hyperplane, say H0, for which b is not parallel to H0. Let a0 ∈ H0 ∩ ∂K(⊂ ∂S). There exists x ∈ K and > 0 such that either b = a0 − x or b = x − a0. In the first case, by convexity x ∈ S, x + b ∈ ∂S ⇒ a0 + b = x + (1 + )b 6∈ S. In the second case, take another supporting hyperplane to S, say, H1, with b not parallel to H1. As … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[4]

L. Bos, S. Ma‘u, and S. Waldron. Extremal growth of polynomials. Anal. Math., 46(2):195–224, 2020

work page 2020
[6]

Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Exterior Monge-Amp` ere solutions.Adv. Math., 222:331–358, 2009

work page 2009
[1]

M. Baran. Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in Rn. Michigan Math. J. , 39(3):395–404, 1992

work page 1992
[2]

Bedford and B

E. Bedford and B. A. Taylor. The Dirichlet problem for a complex Monge- Amp` ere equation.Invent. Math., 37:1–44, 1976

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Bloom, N

T. Bloom, N. Levenberg, and S. Ma‘u. Robin functions and extremal functions. Ann. Pol. Math. , 80:55–84, 2003

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Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Pluripotential theory for convex bodies in Rn. Math. Z., 250:91–111, 2005

work page 2005
[7]

Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Extremal functions for real convex bodies. Ark. Mat., 53(2):203–236, 2015

work page 2015
[8]

Burns, N

D. Burns, N. Levenberg, S. Ma‘u, and S. Revesz. Monge-Amp` ere measures for convex bodies and Bernstein-Markov type inequalities. Trans. Amer. Math. Soc., 362(12):6325–6340, 2010

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

L. Bos, S. Ma‘u, and S. Waldron. Extremal growth of polynomials. Anal. Math., 46(2):195–224, 2020

work page 2020

[2] [6]

Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Exterior Monge-Amp` ere solutions.Adv. Math., 222:331–358, 2009

work page 2009

[3] [1]

M. Baran. Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in Rn. Michigan Math. J. , 39(3):395–404, 1992

work page 1992

[4] [2]

Bedford and B

E. Bedford and B. A. Taylor. The Dirichlet problem for a complex Monge- Amp` ere equation.Invent. Math., 37:1–44, 1976

work page 1976

[5] [3]

Bloom, N

T. Bloom, N. Levenberg, and S. Ma‘u. Robin functions and extremal functions. Ann. Pol. Math. , 80:55–84, 2003

work page 2003

[6] [5]

Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Pluripotential theory for convex bodies in Rn. Math. Z., 250:91–111, 2005

work page 2005

[7] [7]

Burns, N

D. Burns, N. Levenberg, and S. Ma‘u. Extremal functions for real convex bodies. Ark. Mat., 53(2):203–236, 2015

work page 2015

[8] [8]

Burns, N

D. Burns, N. Levenberg, S. Ma‘u, and S. Revesz. Monge-Amp` ere measures for convex bodies and Bernstein-Markov type inequalities. Trans. Amer. Math. Soc., 362(12):6325–6340, 2010

work page 2010