Today I will finally fulfill my earlier promise to revisit the geometry of curves of constant width. I doubt anyone was going to hold me to this promise, but it’s generally good to keep your promises even if you only made them to your own blog. In any event, I have two goals in this post:

- Prove that if two curves have the same constant width then they have the same length (perimeter).
- Prove that among all curves with a given constant width the circle encloses the largest volume.

Let us begin by providing some precise definitions. Recall that a *plane curve* is simply a continuous function , and a plane curve is *closed* if it begins and ends at the same point, i.e. . A closed curve is *simple* if it intersects itself only at the endpoints, meaning only if or and are both endpoints of the interval . The most basic fact about simple closed curves is that they divide the plane into two disconnected regions: a bounded piece (the “inside”) and an unbounded piece (the “outside”). This is called the *Jordan curve theorem*, and as far as I know the simplest proofs use some reasonably sophisticated ideas in algebraic topology (though only a mathematician would think it even needs to be proved!)

Given a simple closed curve , let denote the image of , i.e. the set of all points in the plane that passes through, and let denote together with the points “inside” . A line in the plane is said to be a *supporting line* for if it intersects but does not pass through any interior points of . The set is closed and bounded, so there are exactly two distinct supporting lines for in any given direction. The set of directions in the plane can be parametrized by an angle between and (with the understanding that and represent the same direction). Thus we define a “width” function on the set of directions by letting denote the distance between the supporting lines for in the direction . Here’s what the width looks like in an example:

Finally, we say that has *constant width* if is constant. The goal is to prove that any two curves of constant width have the same length, and that among all curves of constant width the circle of diameter has the largest area. Before proceeding, we need to understand the geometry of constant width curves a little better.

Specifically, we want to show that every curve of constant width is *convex*, meaning contains the line segment between any two of its points. In fact we will prove something a bit stronger: is *strictly convex*, meaning it is convex and contains no line segments (so that the line segment joining any two points in actually lies in the interior of ). This requires a nice little trick that I couldn’t figure out on my own; special thanks to Ian Agol for helping me out on mathoverflow.

**Proposition:** Every curve of constant width is strictly convex.

*Proof:* Let denote the *convex hull* of ; this is by definition the smallest convex set which contains . According to a general fact from convex geometry, the boundary of consists only of points in the boundary of and possibly line segments joining points in the boundary of . So we will show that the boundary of contains no line segments, implying that and hence that is strictly convex.

According to another general fact from convex geometry the supporting lines for are precisely the same as the supporting lines for , and hence has the same constant width as . So assume that the boundary of contains a line segment joining two points and . Since is convex, the line passing through and is a supporting line for . There is exactly one other supporting line for parallel to this line; let denote a point where it intersects . Consider the triangle ; its height is precisely , the width of , so we have that is strictly smaller than at least one of or . Assume and consider the supporting lines for which are perpendicular to the line segment joining and . The points and must lie between (or possibly on) these supporting lines, but the distance between the supporting lines is since has constant width. We conclude that , a contradiction.

*QED*

The reason why strict convexity is important to us is that lines intersect strictly convex curves in a very predictable way:

**Lemma:** Let be a closed strictly convex curve and let be a line which intersects . Then intersects exactly once if it is a supporting line or exactly twice if it is not.

*Proof:* Note that the intersection of two convex sets is again convex, so the intersection is a convex subset of a line. Since is closed and bounded the same must be true of the intersection, so the only possibility is that is a closed interval with . Note that interior points of correspond to interior points of and the boundary points and correspond to boundary points of , so we have that if and only if is a supporting line and otherwise. Thus supporting lines intersect exactly once and any other line which intersects does so exactly twice.

*QED*

We are now ready to calculate the length of a constant width curve. Our strategy is to use the main result of my previous post, “The Mathematics of Throwing Noodles at Paper.” There we saw that if one randomly tosses a curve of length at a lined sheet of paper with line spacing then the expected number of line intersections is given by . So let us toss our curve of constant width at a lined sheet of paper with line spacing . The curve must intersect at least one line and it can’t intersect three or more lines, so it either intersects exactly one line or exactly two lines. The curve intersects exactly two lines if and only if they are supporting lines, and hence each line intersects the curve exactly once by the lemma above. If the curve intersects exactly one line then it cannot be a supporting line and thus the lemma implies that the curve intersects the line exactly twice. In either case the total number of intersections is exactly , and thus the expected number of intersections is . Therefore

and hence . Thus every curve of constant width has length , an assertion consistent at least with the circle of diameter . The result is called *Barbier’s Theorem*, and it has a variety of different proofs; I find the argument using geometric probability to be the most beautiful.

We have now settled the length question; what about area? In fact, to place an upper bound on the area inside a constant width curve we will simply use our length calculation together with the following landmark theorem in geometry:

**Theorem:** Let be the length of a simple closed curve in the plane and let $A$ be the area that it encloses. Then:

with equality if and only if the curve is a circle.

In other words, among all curves with a given length the circle is the unique curve which encloses the largest area. This theorem is called the *isoperimetric inequality*, and it has many beautiful proofs, generalizations, and applications. Our claim about the area enclosed by constant width curves is an immediate corollary since they all have the same length (given a fixed width). I originally intended to prove the isoperimetric inequality in this post using geometric probability, but I would need to take some time to explain how to calculate area probabilistically and I think the post is long enough as it is. Perhaps I will revisit this in the future.