The concurrent normals conjecture

[2024-08-19 Mon]

The concurrent normals conjecture (Problem A3 in Croft, Falconer and Guy "Unsolved problems in geometry") is a basic statement in Euclidean geometry which remains open in dimensions \(\geq 5\) despite the best efforts of mathematicians. Let \(C\) be a convex body in \(\mathbb{R}^n\). The claim is that there exists a point \(q\in C\) which lies on at least \(2n\) (oriented) lines which intersect the boundary orthogonally (i.e. at right-angles). This is certainly true for a generic ellipsoid: the axes all pass through the origin and intersect the boundary orthogonally, and there are \(2n\) of them if you remember their orientations. For a sphere or non-generic ellipsoid, there are points (like the origin) which lie on infinitely many boundary normals.

The conjecture is known to hold in dimensions 2 (where it's easy), 3 (where it was proved by Heil) and 4 (where it was proved by Pardon).

The conjecture can be recast in terms of Morse theory, and that's how Heil and Pardon made progress. Each point \(q\in C\) defines a function \(\varphi_q\colon\partial C\to \mathbb{R}\) on the (spherical) boundary of \(C\), given by \(\varphi_q(p)=|p-q|^2\). The critical points of this function correspond precisely to the boundary normals \(\overrightarrow{pq}\) passing through \(q\). The idea is that the family of functions \(\{\varphi_q\,:\,q\in\partial C\}\) should fail to be nullhomotopic in the closure of the space of Morse functions with fewer than \(2n\) critical points.

I've had some fun bashing my head against this conjecture, and wanted to share with you some of the ways in which I have failed to solve it.

Hurwitz's theorem

If we are willing to ignore the condition that the point \(q\) lies in \(C\), then one proof for \(n=2\) goes like this. Take the Fourier expansion of the support function \(h_C\). The lowest order contribution has the form \(\langle q_1,-\rangle\) for some \(q_1\in\mathbb{R}^n\) because the first order terms in the Fourier expansion are the restrictions of linear functions to the circle. Therefore the lowest term in the Fourier expansion of \(h_{C-q_1}\) is a second harmonic and the same is true of \(\frac{dh_{C-q_1}}{d\theta}\). A theorem of Hurwitz asserts that if a function (in this case \(\frac{dh_{C-q_1}}{d\theta}\)) has vanishing first harmonics then it has at least four zeros.

One might be tempted to try something similar in higher dimensions: expand the support function in terms of spherical harmonics, translate \(C\) to kill the first harmonics and try to appeal to a higher-dimensional version of Hurwitz's theorem. However, the analogous statement for critical points of functions on higher-dimensional spheres (that a function on \(S^{n-1}\) with vanishing first harmonics must have at least \(2n\) critical points) is false; probably this is well-known to experts, but we give an explicit counterexample here because we could find this issue alluded to only obliquely in the literature (e.g. Arnold [Section 12, p.39] ``It is well known that direct attempts to carry over theorems on zeros of linear combinations of eigenfunctions to the multidimensional case are abortive (see, for example `Herman's theorem' in [Courant-Hilbert])''.). The following cubic function restricted to the 2-sphere

\begin{gather*} -0.14475812776097205x^3 + 0.9525264212673581x^2y +\\ -0.3176273850507515x^2z + 0.9326352733462836xy^2 +\\ 0.54832579503517xyz + 0.006426096708431979xz^2 +\\ 1.4333329050246486y^3 - 0.5881612035050371y^2z +\\ 0.6481822137829978yz^2 + 0.31115367318331705z^3 +\\ -0.35387234179093563x^2 + 0.20122198067609898xy +\\ 0.7218185489388442xz - 0.5991576189969507y^2 +\\ 0.2488113937754182yz - 0.45311898819700297z^2 \end{gather*}

has vanishing first harmonics (no linear term) but only four critical points. One can find similar examples by randomly picking the coefficients and using certified root finding algorithms like HomotopyContinuation.jl to locate the critical points. Whilst certified algorithms like this might in principle miss some solutions, in this situation we can show that it has not missed any. Indeed, there is an upper bound on the number of complex solutions (from B\'{e}zout's theorem) and the certified algorithm finds this number of complex solutions in total but only four are real.

Here is the Julia code: 


using Random, HomotopyContinuation, Counters, DynamicPolynomials

function get_random_poly(d::Int,
                         vs::Vector{DynamicPolynomials.PolyVar{true}})
    f = 0
    mons = monomials(vs,2:d)
    for mon in mons
        f += randn(Float64)*mon
    end
    return f
end 

function get_random_eq(d::Int,
                       vs)
    f = get_random_poly(d,[x,y,z])
    return solv_eq(f, [x,y,z,l])
end

function test(N::Int,
              d::Int,
              vs)
    full_list = [get_random_eq(d, vs) for k in 1:N]
    full_nums = [f[2] for f in full_list]
    cept = [f[1] for f in full_list if f[2] < 2*(length(vs)-1)]
    return cept, counter(full_nums)
end

function solv_eq(f, vs)
    nvs = length(vs)
    g = f - vs[1]*(sum([w^2 for w in vs[2:nvs]])-1)
    eqs = [differentiate(g, v) for v in vs]
    sy = System(eqs)
    ss_1 = solve(sy; start_system = :total_degree)
    cert_res = certify(sy, ss_1)
    return f, ndistinct_real_certified(cert_res)
end

function detail(f, vs)
    nvs = length(vs)
    g = f - vs[1]*(sum([w^2 for w in vs[2:nvs]])-1)
    eqs = [differentiate(g, v) for v in vs]
    sy = System(eqs)
    print(total_degree(sy))
    ss_1 = solve(sy; start_system = :total_degree)
    cert_res = certify(sy, ss_1)
    return cert_res
end

function polyh(f, vs)
    nvs = length(vs)
    g = f - vs[1]*(sum([w^2 for w in vs[2:nvs]])-1)
    eqs = [differentiate(g, v) for v in vs]
    sy = System(eqs)
    tracker, starts = polyhedral(sy; only_non_zero = false)
    return count(is_success, track.(tracker, starts))
end

Geometric flows

One could try to ``flow'' the boundary of \(C\) using a parabolic geometric flow. For example, the affine normal flow [Andrews], heat flow of the support function, or Lagrangian mean curvature flow of \(L_{\partial C}\) in \(T^*S^{n-1}\). This would give a family of Lagrangians \(L_t\), and if one could show that the number of intersections \(L_t\cap L_q\) was always decreasing, and that \(L_t\) converged to \(L_\Sigma\) for an ellipsoid \(\Sigma\) (as indeed holds for affine normal flow), then one could deduce the conjecture. However, John Pardon pointed out to us that already with the heat flow of functions on \(\mathbb{R}^2\), cancelling pairs of critical points can appear along the flow (see his MathOverflow post).

References

Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.