Singular set of optimal bodies in Newton's least resistance problem: Plakhov's ridge-symmetry conjecture, conical singular points, and non-disk base domains

Newton's least resistance problem in the concave formulation of Buttazzo and Kawohl (1993) — minimize $\iint_\Omega(1+|\nabla u|^2)^{-1}\,dx$ over concave $u:\Omega\to[0,M]$ — is solved only in Newton's original radially symmetric subcase, and a minimizer is known to be uniquely determined by its singular set, so characterizing that set is the structural core of the problem. The source (Plakhov, J. Dyn. Control Syst. 29, 2022) proved the first necessary conditions for ridge singular points of an optimal body and explicitly left open three linked questions that together constitute the singular-set characterization problem: (1) Conjecture 1 — for the unit disk with upper level set of nonempty interior, all singular points outside the upper level set are ridge points with edge angle $\theta\ne0$ and symmetric face angles $\varphi_1=-\varphi_2$; (2) necessary conditions and admissible geometry for conical singular points of an optimal body; (3) the singular structure when the upper level set is a line segment or the base domain is not a disk. No proof or disproof of any component is currently known; the recent works of Buttazzo (2025) and Wachsmuth (2025) and the conical-part non-optimality theorem of Lokutsievskiy–Wachsmuth–Zelikin give only partial adjacent evidence. The problem is open.

Background

Newton's problem of least resistance, posed by Isaac Newton in 1687 for rotationally symmetric bodies and extended in 1993 by Buttazzo and Kawohl to general convex bodies, asks for the shape of a convex body of prescribed base and height that experiences the least drag when moving at constant velocity through a rarefied medium of non-interacting particles at rest, with perfectly elastic particle–body collisions. In the modern formulation one minimizes the resistance functional

$$F(u)=\iint_\Omega\frac{dx_1\,dx_2}{1+|\nabla u(x_1,x_2)|^2}$$

over the class $\mathfrak{C}_M$ of concave functions $u:\Omega\to[0,M]$, where $\Omega\subset\mathbb{R}^2$ is a compact convex set with nonempty interior (the base of the body) and $M>0$ is its height; the body is the convex compact set $C_u=\{(x,z):x\in\Omega,\ 0\le z\le u(x)\}$ bounded below by the base $\Omega\times\{0\}$ and above by the graph of $u$. Newton solved the problem only in the smaller radially symmetric class; in the full concave class it remains unsolved.

Several facts are known. A minimizer exists (Buttazzo–Ferone–Kawohl 1995, via the theory of nonconvex integrals); every minimizer vanishes on $\partial\Omega$ (Plakhov 2020); at every regular point the gradient satisfies $|\nabla u|\ge1$ or $\nabla u=0$ (Buttazzo–Ferone–Kawohl 1995); wherever a minimizer is twice differentiable its Hessian is degenerate (Brock–Ferone–Kawohl 1996), and in fact wherever it is continuously differentiable its graph contains no extreme points of $C_u$ (points that are not interior points of any segment contained in the body), so the regular part of the optimal surface is developable — a union of line segments — by Plakhov's 'nose stretching' variation technique (2021), which excludes extreme points from any open region where the optimal function is $C^1$; over a disk the minimizer is not radially symmetric (Brock–Ferone–Kawohl 1996); and if the top set has nonempty interior, the slope tends to 1 at almost every boundary point of it (Plakhov 2021). Crucially, a solution is uniquely determined by the closure of its set of singular points (Plakhov 2022), so characterizing the singular set amounts to characterizing the optimal shape itself.

The classification of boundary points of a convex body by the tangent cone (the smallest closed cone with vertex at the point that contains the body) is classical in the extrinsic geometry of convex surfaces: a boundary point is regular if the tangent cone is a half-space, a ridge point if the tangent cone degenerates into a dihedral angle, and a conical point if the tangent cone is nondegenerate, that is, a genuine three-dimensional cone containing no straight line. A singular point is a non-regular boundary point. At a ridge point of the optimal body, the dihedral angle is described by a triple of angles $(\theta,\varphi_1,\varphi_2)$: $\theta\in[0,\pi/2)$ is the angle between the edge of the dihedral angle and the base plane, and $\varphi_1,\varphi_2$ are the angles made by the outward normals of the two faces with a fixed upward direction in the plane orthogonal to the edge, with $-\pi/2<\varphi_2<\varphi_1<\pi/2$. The triple determines the dihedral angle up to motions of the base plane.

The source paper (Plakhov 2022) solves a family of auxiliary two-dimensional least-resistance problems for the surface area measure of a convex body (the measure on the unit sphere obtained by pushing the boundary surface measure onto outward normals) and derives the first necessary conditions for ridge points of an optimal body: (a) if $\theta\in[\pi/4,\pi/2)$ then $2\varphi_1+\varphi_2\le\pi/2$ and $\varphi_1+2\varphi_2\ge-\pi/2$; (b) if $\theta\in(0,\pi/4)$ then additionally $|\varphi_1|\ge\varphi_*$ and $|\varphi_2|\ge\varphi_*$, where $\varphi_*=\arccos\big((1/\sqrt{2})/\cos\theta\big)$; (c) if $\theta=0$ (horizontal edge) then $(\varphi_1,\varphi_2)$ is either $(\pi/4,0)$, or $(0,-\pi/4)$, or satisfies $2\varphi_1+\varphi_2\le\pi/2$, $\varphi_1+2\varphi_2\ge-\pi/2$, $\varphi_1\ge\pi/4$, $\varphi_2\le-\pi/4$. A corollary: all ridge points with horizontal edge lie on the boundary of the top face and of the base. Numerical solutions of Wachsmuth (2014) for the unit disk indicate that for heights $M$ up to about 1.5 the upper level set (ULS) $L=\{x:u(x)=M\}$ — the projection of the flat top face — is a regular polygon with nonempty interior, and the lateral surface carries singular curves joining the polygon's vertices to the boundary of the base, each point of these curves being a ridge point with $\theta\ne0$ and $\varphi_1=-\varphi_2$; this computation is what motivates the conjecture at the heart of the present problem. Complementary results of Lokutsievskiy–Wachsmuth–Zelikin (2022) show that certain conical surface pieces (the convex hull of a vertex lying above the interior of the base and a circular arc of the base boundary, satisfying an explicit inequality) cannot occur on an optimal body, which disproves several families of previously conjectured solutions; refined numerics (Wachsmuth 2025) show dihedral symmetry $D_k$ (the symmetry group of a regular $k$-gon) with $k$ depending on $M$, and extremal points lying on several curves — conjecturally, on two extremal arcs, one of which lies in the plane of symmetry. None of this, however, determines the singular set of an optimal body.

Problem Statement

Let $\Omega\subset\mathbb{R}^2$ be a compact convex set with nonempty interior, let $M>0$, and let $u$ minimize $F(u)=\iint_\Omega(1+|\nabla u|^2)^{-1}\,dx_1dx_2$ over the class of concave $u:\Omega\to[0,M]$; let $C_u=\{(x,z):x\in\Omega,\ 0\le z\le u(x)\}$ be the optimal convex body. Determine the structure of the singular points of $C_u$:

(1) Conjecture 1 of the source paper (its 'unit circle' is the unit disk): if $\Omega$ is the unit disk and the upper level set $L=\{x:u(x)=M\}$ has nonempty interior, prove or disprove that all singular points outside the ULS are ridge points and satisfy $\theta\ne0$ and $\varphi_1=-\varphi_2$ — in particular, that no conical singular points occur off the top face and that every off-top ridge dihedral angle is symmetric.

(2) Establish necessary conditions and the admissible geometry for conical singular points of an optimal body — points of $\partial C_u$ whose tangent cone is nondegenerate, i.e. contains no straight line — analogous to the source paper's necessary conditions for ridge points; the source paper explicitly postpones this study to future work.

(3) Determine which dihedral-angle triples $(\theta,\varphi_1,\varphi_2)$ and which singular curves can occur on the lateral surface when the ULS is a line segment, and when the base domain $\Omega$ is not a disk — regimes the source paper describes as 'unclear'.

These three components share one scientific objective: a complete characterization of the singular set of an optimal body of Newton's least resistance problem in each of its regimes. Because a solution is uniquely determined by its singular set, such a characterization would essentially determine the optimal shape and would settle the structural core of the still-open concave (Buttazzo–Kawohl) formulation. Component (3) is posed for general convex base domains, not merely for special subclasses.

References

1. Alexander Plakhov, On the structure of singular points of a solution to Newton's least resistance problem, Journal of Dynamical and Control Systems 29 (2022) 1161–1174. DOI 10.1007/s10883-022-09616-y, https://doi.org/10.1007/s10883-022-09616-y (arXiv:2203.14235, https://arxiv.org/abs/2203.14235)
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