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Problems of Minimal Resistance and the Kakeya Problem

The Editors of SIAM Review recap the SIGEST paper in the September 2015 issue of SIREV:

The SIGEST paper in this issue, “Problems of Minimal Resistance and the Kakeya Problem,” by Alexander Plakhov, is from the SIAM Journal on Mathematical Analysis.

The paper considers the problem of least resistance. This problem was posed by Newton (whom the author cites) and has been the subject of much recent work. The setting for the simplest instance of the problem is flow of particles striking a solid body. The particles impart a force to the body. The problem is to find the shape that minimizes that resistive force. There are, of course, a few things one must add to the problem to make it tractable. One assumes that the collisions between the particles and the body are elastic, that the particles do not collide with each other, and that the flow is homogeneous. One also limits the types of bodies, generally with a smoothness condition, in the optimization and constrains the class of bodies to eliminate the possibility that a particle can strike the body more than once. The shape is constrained to be a function from a set $\Omega \subset R^2$ with values in [0,1].

The author considers a special case for which one can pose the problem analytically. The formulation normalizes the dimensions of the body so that the minimum value of the functional which maps the surface to its normalized resistance is between 1/2 and 1. In this paper the author shows that the minimum value of the objective function is exactly 1/2, bettering the previous estimate of .582

The proofs are, as you might suspect, quite technical. What the reader may not expect is that the techniques are elementary and little more than multivariable calculus is needed to work through the paper. The author has kindly included some very instructive images which will assist the reader with the geometrical subtleties of the analysis.

Read the paper! (Requires subscription or SIAM membership)

Problems of Minimal Resistance and the Kakeya Problem

SIAM Review, 57(3), 421-434.

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