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arxiv: 1907.05379 · v1 · pith:FWASWJOPnew · submitted 2019-07-11 · 🧮 math.MG · math.CA· math.PR

Analytic and Probabilistic Problems in Discrete Geometry

Gergely Ambrus This is my paper

Pith reviewed 2026-05-24 22:24 UTC · model grok-4.3

classification 🧮 math.MG math.CAmath.PR
keywords polarization problemunit vectorsHilbert spaceconvex chainsrandom points in trianglediscrete geometrylimit shape
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The pith

Any n unit vectors in a Hilbert space admit a unit vector v such that the sum of 1 over squared inner products is at most n squared, and only orthonormal systems are locally extremal in full dimension.

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

The thesis proves a bound for the strong polarization problem: given any sequence of n unit vectors in a real Hilbert space, a unit vector v exists making the sum of reciprocal squared inner products at most n squared. The two-dimensional case follows from complex analytic methods, while a tensorisation result modeled on John's theorem extends the bound to higher dimensions and shows that only orthonormal systems are full-dimensional locally extremal configurations. The same bound is derived for the original polarization problem. In a separate probabilistic setting, n independent uniform points are placed in a triangle; the longest convex chain L_n between two fixed vertices satisfies E[L_n] asymptotically equal to alpha n to the one-third power, with alpha between 1.5 and 3.5, together with concentration inequalities that produce a limit shape result.

Core claim

For any sequence of n unit vectors in a real Hilbert space there exists a unit vector v such that the sum of 1 over inner-product squared is at most n squared; the only full-dimensional locally extremal configurations are orthonormal systems. The expectation of the longest convex chain length L_n satisfies E[L_n] ~ alpha n^{1/3} with 1.5 < alpha < 3.5.

What carries the argument

The tensorisation result analogous to John's theorem, which reduces higher-dimensional local extremality to the two-dimensional case, together with the probabilistic study of maximal convex chains among uniform random points in a triangle.

If this is right

  • The bound holds for the weaker original polarization problem as well.
  • Strong concentration inequalities for L_n imply a limit shape theorem for the longest convex chains.
  • Orthonormal systems realize the polarization bound.
  • The constant alpha is conjectured to equal exactly 3.

Where Pith is reading between the lines

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

  • The tensorisation technique could extend to other vector polarization inequalities beyond the present setting.
  • The concentration and limit-shape methods may apply to longest chains or paths in other random geometric models.
  • Refined tail bounds on convex chains would be needed to decide whether alpha equals 3.

Load-bearing premise

The tensorisation result holds and implies that only orthonormal systems are locally extremal in higher dimensions.

What would settle it

A concrete set of n unit vectors for which every unit vector v yields a sum strictly larger than n squared, or a large-n simulation in which the observed E[L_n] falls outside the interval from 1.5 n^{1/3} to 3.5 n^{1/3}.

Figures

Figures reproduced from arXiv: 1907.05379 by Gergely Ambrus.

Figure 1.1
Figure 1.1. Figure 1.1: 1/Gz(z) and its equioscillating approximation (dotted) It is easy to check that the condition of Proposition 1.14 on the position of the zeroes of g(z) is not necessary. Comparing the coefficients in (1.29) yields that if n is odd, then (1.29) gives a one-to-one relation between g(z) and h(z), while if n is even, say n = 2k, then regardless of the coefficient of z k in g(z), we obtain the same h(z) (in w… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: First three steps of the iteration (plain, dashe [PITH_FULL_IMAGE:figures/full_fig_p029_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: P(H) in the 3-dimensional case Local extremality with respect to Conjecture 1.26 is defined usually. Locally ex￾tremal matrices (and ellipsoids) are characterised via the following theorem; note that the result holds for lower dimensional extremal cases as well. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_1_3.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The special parabola Now, let K ⊂ R 2 be an arbitrary convex body. B´ar´any [14] proved that if we take all the convex polygons spanned by points of Xn, then the majority of them is close to K0. Therefore, K0 is the limit shape of the random convex polygons inside K. In the article [16], strengthening this result, an almost sure limit theorem and a central limit theorem for convex chains are proved. Next… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Characterisation of Γ As we have mentioned earlier, the special parabola arc Γ ⊂ T is characterized by the fact that it has the largest affine length among all convex curves connecting p0 and p2 within T. This is a consequence of the following theorem from [20]. Assume that a line ℓ intersects the sides p0p1 resp. p1p2 at points q0 and q2. Let q be a point on the 55 [PITH_FULL_IMAGE:figures/full_fig_p05… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The parabola Γr In measuring distances from Γ it will be convenient to use the parabolic arcs Γr = {(x, y) ∈ T : √ x + √ y = √ 1 + r}, where r ∈ (−1, 1). Then Γ0 = Γ, and Γr is the homothetic copy of Γ with ratio of homothety 1 + r, and center of homothety at the origin, see [PITH_FULL_IMAGE:figures/full_fig_p069_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Squares above Γ Let the vertices of D in counter-clockwise order be (x, y), (x, y+δ), (x−δ, y+δ) and (x−δ, y). It is easy to see that among the lines ℓ(q) with q ∈ D, the ones with extremal slopes are the ones belonging to (x, y) and (x − δ, y + δ). Let us call p1 = (x, y), the lower right vertex, and p2 = (x− δ, y + δ), the upper left vertex of D, see [PITH_FULL_IMAGE:figures/full_fig_p072_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Covering ℓ(D) with a bow-tie We divide the rest of the argument into two parts depending on the size of ϕ. If ϕ > 0.17, then 1/ √ 2 sin 2ϕ < 1.23 < √ 5 − 1, and hence by (2.27), σ 6 4(√ 5 − 1) δ. Therefore Dl ∪ Dr can be covered with a “bow-tie”: the union of two oppositely placed trapezoids with shorter base length at most √ 2δ, whose angle between the non-parallel opposite sides is less than 4(√ 5 − 1)… view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Estimating the expectations For four points q0 = (0, 1), q1, q2 and q3 = (1, 0) in this order on Γ, denote Si the triangle delimited by the tangents to Γ at qi−1, qi , and by the segment qi−1qi , i = 1, 2, 3; see [PITH_FULL_IMAGE:figures/full_fig_p075_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Results for n −1/3ELn, illustrated as a function of n 1/3 . With the above method, the search can be executed for up to 5 · 104 active points, in which case examining one sample takes about 2 minutes. As the experiments show, this provides a good approximation for n’s up to order 106 . In each experiment, we increased the width of the searched neighbourhood until the increment did not generate a signific… view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the linear interpolation of [PITH_FULL_IMAGE:figures/full_fig_p080_2.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: Distribution of Ln, 500 samples, n = 253 and n = 106 . 80 [PITH_FULL_IMAGE:figures/full_fig_p080_2_8.png] view at source ↗
read the original abstract

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence $u_1,\dots, u_n$ of norm 1 vectors in a real Hilbert space $\mathscr H$, there exists a unit vector $v \in \mathscr H$, such that $$ \sum \frac{1}{\langle u_i, v \rangle^2} \leq n^2. $$ The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take $n$ independent, uniform random points in a triangle $T$. Convex chains between two fixed vertices of $T$ are defined naturally. Let $L_n$ denote the maximal size of a convex chain. We prove that the expectation of $L_n$ is asymptotically $\alpha \, n^{1/3}$, where $\alpha$ is a constant between 1.5 and 3.5 -- we conjecture that the correct value is 3. We also prove strong concentration results for $L_n$, which, in turn, imply a limit shape result for the longest convex chains.

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

1 major / 2 minor

Summary. The manuscript addresses two problems in discrete geometry. The first establishes the strong polarization inequality: for any n unit vectors in a real Hilbert space, there exists a unit vector v such that the sum of 1 over squared inner products is at most n squared. The 2D case uses complex-analytic methods; a tensorisation result analogous to John's theorem on maximal-volume ellipsoids is proved to deduce that orthonormal systems are the only full-dimensional locally extremal configurations (and the same for the weaker polarization problem). The second chapter considers n uniform random points in a triangle and shows that the expected length E[L_n] of the longest convex chain between two fixed vertices satisfies E[L_n] ~ α n^{1/3} for a constant α with 1.5 < α < 3.5 (conjectured to be 3), together with strong concentration and a limit-shape result.

Significance. If the results hold, the polarization theorem supplies a sharp, dimension-independent bound together with a clean characterization of extremals, which would be of interest in geometric inequalities and functional analysis. The probabilistic-geometry results give explicit asymptotic bounds, concentration, and a limit shape for convex chains; the provision of concrete numerical bounds on α and an explicit conjecture adds falsifiability. The tensorisation argument, if fully rigorous, would be a reusable technique.

major comments (1)
  1. [Chapter 1] Chapter 1 (tensorisation step): the deduction that only orthonormal systems are full-dimensional locally extremal configurations rests on a tensorisation result analogous to John's theorem. The manuscript must supply a self-contained verification that this tensorisation applies directly to the functional ∑ 1/⟨u_i, v⟩² (or its logarithm) without extra assumptions on the configuration or the functional; the analogy alone does not automatically guarantee that local extremality transfers from the 2D case.
minor comments (2)
  1. [Abstract] Abstract: the notation 'asymptotically α n^{1/3}' should be replaced by a precise statement (e.g., E[L_n]/n^{1/3} → α in probability or in expectation) to avoid ambiguity.
  2. [Chapter 2] The interval 1.5 < α < 3.5 is wide; if the proof yields explicit constants, a brief indication of how the lower and upper bounds are obtained would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for identifying a point that requires clarification in Chapter 1. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Chapter 1] Chapter 1 (tensorisation step): the deduction that only orthonormal systems are full-dimensional locally extremal configurations rests on a tensorisation result analogous to John's theorem. The manuscript must supply a self-contained verification that this tensorisation applies directly to the functional ∑ 1/⟨u_i, v⟩² (or its logarithm) without extra assumptions on the configuration or the functional; the analogy alone does not automatically guarantee that local extremality transfers from the 2D case.

    Authors: We agree that the current exposition would benefit from a more explicit, self-contained argument showing that the tensorisation result applies directly to the functional F(v) = ∑ 1/⟨u_i, v⟩² (or log F) without additional hypotheses. While the manuscript states that a tensorisation result analogous to John's theorem is proved and then used to transfer local extremality from the planar case, the transfer step is not written out in full detail for this specific functional. In the revised version we will insert a dedicated subsection that (i) states the tensorisation lemma with the precise hypotheses needed for F, (ii) verifies that those hypotheses hold for any finite collection of unit vectors, and (iii) shows how a local extremal configuration in higher dimension must reduce to an orthonormal system by iterating the 2-D case. This will make the argument independent of the analogy alone. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on independent analytic and probabilistic proofs.

full rationale

The abstract states that the 2D polarization case is proved via complex analytic methods, a tensorisation result (analogous to John's theorem) is proved to characterize higher-dimensional extremal systems, and the convex-chain expectation bounds plus concentration are proved directly. No quoted equations or steps reduce any claim to a fitted parameter renamed as prediction, a self-definition, or a load-bearing self-citation whose validity is presupposed; the central results are presented as derived from external tools within the work itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or fitted in the abstract; the claims rest on standard analytic and probabilistic techniques applied to the stated problems.

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Aldous, P

    D. Aldous, P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. 36 (1999), 413–432

    work page 1999

  2. [2]

    N. Alon, J. Spencer, The probabilistic method. 2nd ed. John Wiley & Sons, New York, 2000

    work page 2000

  3. [3]

    Ambrus, I

    G. Ambrus, I. B´ ar´ any,Longest convex chains. Random Structures and Algorithms 35 (2009), no. 2, 137–162

    work page 2009

  4. [4]

    Anagnostopoulos and Sz

    V. Anagnostopoulos and Sz. R´ ev´ esz,Polarization constants for products of linear functionals over R2 and C2 and the Chebyshev constants of the unit sphere. Publ. Math. Debrecen, 68 (2006), 63–75

    work page 2006

  5. [5]

    G. E. Andrews, Asymptotic expression for the number of solutions of a gener al class of Diophantine equations. Trans. Amer. Math. Soc., 99 (1961), 272–277

    work page 1961

  6. [6]

    Arias-de-Reyna, Gaussian variables, polynomials and permanents

    J. Arias-de-Reyna, Gaussian variables, polynomials and permanents. Linear Al- gebra Appl. 285 (1998), 395–408

    work page 1998

  7. [7]

    J. Baik, P. A. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations . J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178

    work page 1999

  8. [8]

    K. M. Ball, The plank problem for symmetric bodies , Invent. Math. 104 (1991), 535–543

    work page 1991

  9. [9]

    K. M. Ball, An elementary introduction to modern convex geometry. In: Flavors of Geometry (Silvio Levy ed.), MSRI lecture notes 31, Cambridge Univ. Press (1997), 1–58

    work page 1997

  10. [10]

    K. M. Ball, Convex Geometry and Functional Analysis , in: Handbook of the geometry of Banach spaces, vol. 1 (ed. W. B. Johnson and J. Lin denstrauss) , Elsevier (2001), 161–194

    work page 2001

  11. [11]

    K. M. Ball, The complex plank problem , Bull. London Math. Soc. 33 (2001), 433–442

    work page 2001

  12. [12]

    K. M. Ball, M. Prodromou, A Sharp Combinatorial Version of Vaaler’s Theorem. Bull. Lond. Math. Soc. 41 (2009), 853–858

    work page 2009

  13. [13]

    B´ ar´ any,Affine perimeter and limit shape

    I. B´ ar´ any,Affine perimeter and limit shape. R. reine angew. Math 484 (1997), 71–84

    work page 1997

  14. [14]

    B´ ar´ any,Sylvester’s question: the probability that n points are in convex position

    I. B´ ar´ any,Sylvester’s question: the probability that n points are in convex position. Ann. Probab. 27 (1999), no. 4., 2020–2034

    work page 1999

  15. [15]

    B´ ar´ any,Random points and lattice points in convex bodies

    I. B´ ar´ any,Random points and lattice points in convex bodies. Bull. Amer. Math. Soc. 45 (2008), no. 3, 339–365. 81

    work page 2008

  16. [16]

    B´ ar´ any, G

    I. B´ ar´ any, G. Rote, W. Steiger, C.-H. Zhang, A central limit theorem for convex chains in the square. Discrete Comput. Geom. 23 (2000), 35–50

    work page 2000

  17. [17]

    B´ ar´ any, M

    I. B´ ar´ any, M. Prodromou, On maximal convex lattice polygons inscribed in a plane convex set. Israel J. Math. 154 (2006), 337–360

    work page 2006

  18. [18]

    Bastero, M

    J. Bastero, M. Romance, John ’s decomposition of the identity in the non-convex case. Positivity 6 (2002), 1–16

    work page 2002

  19. [19]

    Ben ´ ıtez, Y

    C. Ben ´ ıtez, Y. Sarantopoulos and A. M. Tonge, Lower bounds for norms of prod- ucts of polynomials. Math. Proc. Camb. Phil. Soc. 124 (1998), 395–408

    work page 1998

  20. [20]

    Blaschke, Vorlesungen ¨Uber Differenzialgeometrie II

    W. Blaschke, Vorlesungen ¨Uber Differenzialgeometrie II. Affine Differenzialge- ometrie. Springer, Berlin, 1923

    work page 1923

  21. [21]

    Dineen, Complex Analysis on Infinite Dimensional Spaces

    S. Dineen, Complex Analysis on Infinite Dimensional Spaces. Springer Mono- graphs in Mathematics, Springer-Verlag, Berlin, 1999

    work page 1999

  22. [22]

    Enriquez, Convex chains in Z2

    N. Enriquez, Convex chains in Z2. To appear. Preprint available online at http://arxiv.org/abs/math.PR/0612770

    work page arXiv

  23. [23]

    Erd´ elyi, Extremal properties of polynomials

    T. Erd´ elyi, Extremal properties of polynomials. In: A Panorama of Hungarian Mathematics in the 20 th Century, J´ anos Horv´ ath (Ed.), Springer Verlag, New York, 2005, 119–156

    work page 2005

  24. [24]

    P. E. Frenkel, Pfaffians, hafnians and products of real linear functionals. Math. Res. Lett. 15 (2008), no. 2., 351–358

    work page 2008

  25. [25]

    Glader and G

    C. Glader and G. H¨ ogn¨ as,An equioscillation characterization of finite Blaschke products, Complex Variables 42 (2000), 107–118

    work page 2000

  26. [26]

    John, Extremum problems with inequalities as subsidiary conditi ons

    F. John, Extremum problems with inequalities as subsidiary conditi ons. Studies and essays presented to R. Courant on his 60 th birthday, Interscience, New York (1948), 187–204

    work page 1948

  27. [27]

    Lax, Proof of a conjecture of P

    P. Lax, Proof of a conjecture of P. Erd˝ os on the derivative of a polyn omial, Bull. Amer. Math. Soc. 50 (1944), 509–513

    work page 1944

  28. [28]

    Leung, W

    Y. Leung, W. Li, Rakesh, The d th linear polarization constant of Rd. Journ. Funct. Anal. 255 (2008), 2861–2871

    work page 2008

  29. [29]

    B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux , Adv. Math. 26 (1977), 206–222

    work page 1977

  30. [30]

    Matolcsi, The linear polarization constant of Rn

    M. Matolcsi, The linear polarization constant of Rn. Acta Math. Hungar. 108 (2005), no. 1–2, 129–136

    work page 2005

  31. [31]

    Matolcsi, A geometric estimate on the norm of product of functionals

    M. Matolcsi, A geometric estimate on the norm of product of functionals. Linear Algebra Appl. 405 (2005), 304–310

    work page 2005

  32. [32]

    Matolcsi and G

    M. Matolcsi and G. Mu˜ noz, On the real polarization problem. Math. Inequal. Appl. 9 (2006), 485–494. 82

    work page 2006

  33. [33]

    Pappas, Sz

    A. Pappas, Sz. R´ ev´ esz,Linear polarization constants of Hilbert spaces. J. Math. Anal. Appl. 300 (2004), 129–146

    work page 2004

  34. [34]

    P´ olya, G

    Gy. P´ olya, G. Szeg˝ o,Problems and Theorems in Analysis. Springer-Verlag, New York-Heidelberg, 1976. (Original: Aufgaben und Lehrs¨ atz e aus der Analysis, Springer, Berlin, 1925.)

    work page 1976

  35. [35]

    R´ enyi, R

    A. R´ enyi, R. Sulanke, ¨Uber die konvexe H¨ ulle vonn zuf¨ allig gew¨ ahlten Punkten. Z. Wahrsch. Verw. Gebiete 2 (1963), 75–84

    work page 1963

  36. [36]

    R´ ev´ esz and Y

    Sz. R´ ev´ esz and Y. Sarantopoulos, Plank problems, polarization, and Chebyshev constants. J. Korean Math. Soc., 41 (2004) no. 1, 157–174

    work page 2004

  37. [37]

    Riesz, Formule d’interpolation pour la d´ eriv´ ee d’un polynˆ ome trigonom´ etrique, C

    M. Riesz, Formule d’interpolation pour la d´ eriv´ ee d’un polynˆ ome trigonom´ etrique, C. R. Acad. Sci. Paris 158 (1914), 1152–1154

    work page 1914

  38. [38]

    Ryan and B

    R. Ryan and B. Turett, Geometry of Spaces of Polynomials. J. Math. Anal. Appl. 221 (1998), 698–711

    work page 1998

  39. [39]

    Szeg˝ o, Orthogonal polynomials

    G. Szeg˝ o, Orthogonal polynomials. AMS Colloquium Publications, New York, 1939

    work page 1939

  40. [40]

    Talagrand, A new look at independence

    M. Talagrand, A new look at independence. Ann. Probab. 24 (1996), 1–34

    work page 1996

  41. [41]

    C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel . Comm. Math. Phys., 159 (1994), 151–174

    work page 1994

  42. [42]

    Valtr, The probability that n points are in convex position

    P. Valtr, The probability that n points are in convex position. Discrete Comput. Geom. 13 (1995), 637–643

    work page 1995

  43. [43]

    A. M. Vershik and S. V. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables. Dokl. Acad. Nauk. SSSR, 233 (1977), 1024–1027. 83

    work page 1977