If we describe the surface of the body
as the graph of a function *u*, then we can state the
problem in mathematical terms in the following way:

Newton himself provided a solution to this minimization problem in the case where Omega is a ball in R², limiting himself to functions that have radial symmetry. This reduces the 2-D integral to a one-dimensional one and greatly simplifies the analysis. Here are some pictures of the solution that he found:

The left picture is for M = 2, the right picture for M = 0.5. In both cases the radius of the ball is 1. (Thanks to Paolo Guasoni for the pictures). Although it is counterintuitive, the solution provided by Newton contains a flat zone at height M which varies in size (depending on M) but is always present.

Newton's formula for the air resistance (the integral in (1)) is found in many engineering handbooks, and his solution is presented as a simple application of the calculus of variations. Throughout the three hundred years that have passed since the first publication of the Principia, no-one seemd to ask a very simple and obvious question: if Omega is a ball, is the minimizer necessarily radially symmetric? In other words, does the solution copy the symmetry of the domain? And was Newton therefore justified in assuming that the minimizer is radially symmetric?

Intuition tells us that this is true, as does a comparison with many
other minimization problems. But it is false, in fact: in 1996 a paper
was published by F. Brock, V. Ferone, and B. Kawohl, where it was proved
that the minimizer on the ball is *not* radially symmetric.
This was done by showing that the minimizer could not be at the same time both
strictly concave and smooth. Since Newton's solution has both these
properties - apart from the top flat zone - this shows that this function
does not minimize among all functions. Since it does minimize among radially
symmetric functions, the `real' minimizer must be non-radially symmetric.

This result opened the witch-hunt for the answer of the immediate question:

We managed to prove a couple of things, though.

T. Lachand-Robert and Mark A. Peletier, An Example of Non-convex Minimization and an Application to Newton's Problem of the Body of Least Resistance, Annales de l'Institut Henri Poincaré, Analyse non linéaire 18, pp. 179-198 (2001)

As a first step towards a characterization we
generalized the result mentioned above to functions wth general
regularity, i.e.
*u* must contain a dense set
of straight line segments.

IfOr, said differently, the graph ofuis the minimizer of Newton's problem, then for any open subet of Omega,uis not strictly convex on that subset.

I called this result a generalization, but I mean to stress that the proof is not a simple extension of the proof given by Brock et al. Because of the lack of regularity, choosing an appropriate class of perturbations of the minimizer is highly non-trivial, since most perturbations make the function non-concave and therefore take it outside of the set of admissible functions.

The fact that *u* (the minimizer) is non-strictly
concave does not give a lot information about its actual form, however.
Whilst trying to imagine possible forms, it might be useful to think of
the function here on the right, discovered by Guasoni
(he called it the screwdriver). For large values of M (M >= 2)
the screwdriver has a lower resistance than Newton's function. Note that
the screwdriver is indeed non-strictly concave, everywhere: from
every point on the circular base there runs at least one straight
line up to the top.

T. Lachand-Robert and Mark A. Peletier, Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions (930 kB), Mathematische Nachrichten, Vol. 226, pp. 153-176 (2001).

One way of describing the screwdriver is as the convex hull, or envelope,
of two sets in 3-space: the circular base and the line segment at the top.
In this paper we considered
the following question: *supposing* the minimizer is
of this form, then what should it look like? In this class the functions
are fully characterized by the form of the top zone, and it turns
out that this set must be a regular polygon. The screwdriver corresponds to
the simplest polygon, with two sides, and it is the optimal one
when M is large; as we decrease M, optimality jumps from one
regular polygon to the next:

T. Lachand-Robert and Mark A. Peletier, Extremal Points of a Functional on the Set of Convex Functions, Proceedings of the AMS, Vol. 127, pp. 1723-1727 (1999).

As a spin-off of the work on the Newton's problem, which involves
a non-convex functional, we also came across
some new results on the minimization of *convex* functionals
on the set of convex (or concave) functions.
Provided the boundary values are restricted this paper gives an
interesting, and surprisingly simple, characterisation of extremal
points of such a functional.

T. Lachand-Robert and Mark A. Peletier, The Minimum of Quadratic Functionals of the Gradient on the Set of Convex Functions (72 kB), Calc. Var. Partial Differential Equations, Vol. 15, pp. 289-297 (2002).

Another spin-off result concerns minimizing sequences for the
minimization problem

under the additional constraint that the functions u are convex. Without the convexity constraint, the minimum is equal to the smallest eigenvalue of the matrix A; under the convexity constraint, however, the situation is different, and depends on the boundary of the set Omega. See the paper itself for the details.

under the additional constraint that the functions u are convex. Without the convexity constraint, the minimum is equal to the smallest eigenvalue of the matrix A; under the convexity constraint, however, the situation is different, and depends on the boundary of the set Omega. See the paper itself for the details.

Some of these results are summarized in the Comptes Rendus paper T. Lachand-Robert and Mark A. Peletier,Minimisation de fonctionnelles dans un ensemble de fonctions convexes, C. R. Acad. Sci. Paris, t. 325, Série I, p. 851-855 (1997).

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