Global minimizers of Newton's problem of minimal resistance in the class of convex bodies and the infinite-height limit

Newton's problem of minimal resistance asks for the convex body of given base (the unit disk) and height $M>0$ experiencing the least resistance when moving through a rarefied medium of non-interacting particles. Minimizers exist in the class $C_M$ of closed convex functions and satisfy strong necessary conditions — attained gradients lie in $\{0\}\cup[1,\infty)$, the body meets its base plane at the disk boundary, no strictly convex smooth patches occur, and minimizers are not radially symmetric — yet their exact shapes are unknown. Lokutsievskiy, Wachsmuth and Zelikin (2022) proved that certain conical parts of the boundary force non-optimality, disproving the conjectured bodies of Lachand-Robert–Peletier (all heights), of Wachsmuth (for $M\in[0.95,1.08]$), and of Lokutsievskiy–Zelikin; the test leaves the heights $M\le 0.94$ and $M\in[1.09,1.4]$ undecided, and for $M>\bar M$ with $\bar M\in(1.4,1.5)$ no structured candidate is known at all. The same work introduced the infinite-height limiting problem $J_\infty(u)=\int_\Omega|\nabla u|^{-2}dx$ over convex $u:\Omega\to[-1,0]$, proved existence and non-radiality of its minimizers and excluded the developable class, while leaving open their uniqueness, qualitative structure, and gradient-magnitude set (numerical limit profiles exhibit gradients below $1$). The problem posed here is to characterize the global minimizers of Newton's problem in the convex class for finite heights and in the infinite-height limit.

Background

Newton's problem of minimal resistance, posed by Isaac Newton in 1687 in the Principia, is one of the oldest problems of the calculus of variations. A convex body whose upper boundary is the flat base $\Omega\times\{0\}$ (here $\Omega$ is the unit disk in $\mathbb{R}^2$) moves in the negative $z$-direction through a rarefied medium: a cloud of non-interacting point particles at rest. If the lower boundary of the body is the graph of a function $u:\Omega\to[-M,0]$, where $M>0$ is the height, each particle hitting the surface transfers momentum to the body. Under the single-impact assumption — each particle meets the body at most once, which is automatic when the body is convex — the total resistance is, up to a constant factor,

$$J(u)=\int_\Omega\frac{dx\,dy}{|\nabla u(x,y)|^2+1}.$$

The admissible class is

$$C_M=\{u:\Omega\to[-M,0]\mid u\ \text{convex and closed}\},$$

where "closed" means lower semicontinuous (a canonical representative of each equivalence class, since $J(u)=J(\mathrm{cl}\,u)$). The literature is split between this convex convention and the equivalent concave convention $v=-u\ge 0$.

Minimizers of $J$ over $C_M$ exist (Buttazzo–Ferone–Kawohl, 1995), and several strong necessary conditions are known. At almost every point of a minimizer, $|\nabla u|\in\{0\}\cup[1,\infty)$: the gradient either vanishes (a flat piece of the boundary) or has magnitude at least $1$ (slope at least $45^\circ$). A minimizer tends to $0$ at $\partial\Omega$ (Plakhov, 2020), contains no open set on which it is $C^2$ and strictly convex, and is not radially symmetric. Newton himself solved the problem within bodies of revolution; his solution has a flat nose joined to a strictly convex lateral part at a $45^\circ$ corner. The belief that Newton's body is optimal among all convex bodies survived three centuries until 1996, when Guasoni exhibited a "screwdriver" body — the convex hull of the base disk and a vertical line segment — with strictly smaller resistance for heights $M\ge 2$ (an analytical proof of the non-optimality of Newton's body is due to Brock–Ferone–Kawohl, 1996). Because of the gradient dichotomy and the absence of smooth strictly convex pieces, the Euler–Lagrange equation is unusable: any minimizer is necessarily non-smooth, and its exact shape has remained unknown.

Three families of structured candidates have been proposed. Lachand-Robert–Peletier (2001) suggested bodies that are the convex hull of the base and a regular polygon at depth $-M$. Wachsmuth (2014), refining earlier numerical solutions of Lachand-Robert–Oudet (2005) that discretize the graph as a convex hull, conjectured that for heights $M\in(0,\bar M)$ with $\bar M\in(1.4,1.5)$ the optimal body is the convex hull of the base and $k\ge 3$ congruent copies of a planar convex curve arranged with dihedral $D_k$ symmetry (the symmetry group of a regular $k$-gon), the curve being determined by a one-dimensional variational problem. Lokutsievskiy–Zelikin (2020) solved the problem within bodies having a vertical mirror plane and developable side boundary (a developable surface is a ruled surface of zero Gaussian curvature), proving local optimality within that restricted class.

Lokutsievskiy–Wachsmuth–Zelikin (2022) made two main contributions. First, a non-optimality test for conical parts — patches of the boundary that lie on a cone whose vertex sits on the bottom face of the body: if such a part satisfies an explicit angular inequality (their Corollary 7), the body cannot be optimal in $C_M$. This test disproves the Lachand-Robert–Peletier bodies for every $M>0$, the Wachsmuth family for $M\in[0.95,1.08]$, and the Lokutsievskiy–Zelikin developable bodies for all tabulated heights. For $M\le 0.94$ and $M\in[1.09,1.4]$ the test does not apply to the best structured candidates, so their global optimality is undecided; for $M>\bar M$ no structured conjecture exists at all. Second, they introduced the infinite-height limit: rescaling $\hat u=u/M$ rewrites the problem as $J_M(u)=\int_\Omega(|\nabla u|^2+M^{-2})^{-1}dx\to\min$ over $\hat C=C_1$, whose formal limit is

$$J_\infty(u)=\int_\Omega\frac{dx\,dy}{|\nabla u(x,y)|^2}\to\min,\qquad u\in\hat C.$$

They proved that $J_\infty$ has minimizers, that every accumulation point of rescaled finite-$M$ minimizers minimizes $J_\infty$, and that no minimizer of $J_\infty$ is radially symmetric: the unique radial minimizer $(x_1^2+x_2^2)^{2/3}-1$ (value $27\pi/32\approx 2.651$) violates the no-strict-convexity condition and is beaten by a simple screwdriver (value $\approx 2.145$), while the best developable candidate ($\approx 2.1402$) carries a non-optimal conical part and is therefore excluded. Numerically computed limit profiles exhibit gradient magnitudes of about $0.9863$ — strictly between $0$ and $1$, in contrast to the finite-$M$ dichotomy — but no rigorous description of the gradient set, symmetry class, or uniqueness of $J_\infty$ minimizers is known.

Problem Statement

Characterize the global minimizers of Newton's problem of minimal resistance in the class of convex bodies with unit-disk base. Concretely, let $\Omega$ be the unit disk in $\mathbb{R}^2$, let $M>0$ be the height, and consider the admissible class

$$C_M=\{u:\Omega\to[-M,0]\mid u\ \text{convex and closed}\},$$

where "closed" means lower semicontinuous, together with the resistance functional

$$J(u)=\int_\Omega\frac{dx\,dy}{|\nabla u(x,y)|^2+1}.$$

1. For each height $M>0$, determine the minimizer(s) of $J$ over $C_M$ — their explicit shape, or a complete qualitative description fixing the symmetry class (e.g. dihedral $D_k$, the symmetry group of a regular $k$-gon), the flat, conical, and developable parts of the boundary (a developable surface being a ruled surface of zero Gaussian curvature), the singular set (the points where the boundary surface fails to be smooth), and the set of attained gradient magnitudes. This includes deciding the global optimality or non-optimality of the structured candidates in the ranges $M\le 0.94$ and $M\in[1.09,1.4]$, where the conical-part test of Lokutsievskiy–Wachsmuth–Zelikin does not apply, and identifying the optimal bodies for the heights $M\in[0.95,1.08]$, where the conjectured structured family is disproved, and $M>\bar M$ with $\bar M\in(1.4,1.5)$, where no structural conjecture exists at all.

2. Characterize the minimizers of the infinite-height limiting problem

$$J_\infty(u)=\int_\Omega\frac{dx\,dy}{|\nabla u(x,y)|^2}$$

over the rescaled class $\hat C=C_1$: uniqueness or multiplicity, qualitative geometric structure, and the set of gradient magnitudes attained — in particular whether $|\nabla u|$ may take values strictly between $0$ and $1$, as numerical computations of the limit profiles suggest.

The two sub-questions form one jointly posed scientific objective: the complete description of the least-resistance convex body as a function of its height, including the limit of infinite height; the two regimes are linked because every accumulation point of rescaled finite-$M$ minimizers minimizes $J_\infty$.

References

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