How wide can a unit-length planar curve be?

The width of a bounded set in the plane is the minimum distance between two parallel lines bounding the set.

Suppose that we have some curve $C: [0,1]\to\mathbb{R}^2$ of unit length. How large can its width be? Equivalently, if we would like to bend a wire into a shape of unit width, how much wire do we need? (I'll use the latter framing in this post because it makes the numbers a little nicer.)

If one tries forming the wire into a circle or semicircle, $\pi$ units of length are needed. All but one of the sides of a unit square only takes $3$ units. Using $270$ degrees of a circular arc one can do better, with length $\frac{3\pi(2-\sqrt{2})}{2}=2.7604\ldots$.

After playing around with different natural options for a bit, the winner appears to be $\frac{4}{\sqrt{3}}\approx2.3094$, given by forming the wire into a $60$-degree angle so that its convex hull is the equilateral triangle of width 1.

However, it turns out that this is not optimal! Consider the following construction: Fix $0.5<x<\frac{\sqrt{3}}{2}$. Start with the triangle formed by the points $(0,1), (-x,0), (x,0)$. Then, add the circular arcs of radius $1$ from each of the latter two points as they pass through the opposite side of the triangle. Take the convex hull of these three points and two circular arcs, and omit the line segment of length $2x$ at the bottom; the resulting set is our wire.

Above is a diagram of this construction: the original triangle is marked in blue, the circular arcs (of which only part make it to the convex hull) are marked in red, the straight-line tangents to said arcs are in black, and the portion of the convex hull not on the wire is marked with a dashed line.

It is fairly easy to see that this construction has width $1$, and that the resulting length is

$$2\left(x+\sqrt{4x^2-1}+\arctan(1/x)-\arctan(x)-\arccos(1/2x)\right)$$

As it turns out, this function is minimized around $x=0.521795$, with a wire length of around $2.2783$. (In fact, the derivative of the above function is a rational function, so we can be more precise: $x$ is the unique positive real root of the polynomial $3x^6 + 9x^4 + x^2 - 1$.)

Is this construction optimal? I do not believe there are any obvious local improvements one can make, but it's possible that a radically different wire shape could improve on this configuration.

Obviously, there is a lower bound of $1$ on the length of any solution, and by considering the width in two orthogonal directions one can increase this to $\sqrt{2}$. It's tempting to say that the length must be at least $2$, because there must be a point at distance $1$ from the midpoint of the wire, but this only follows if one makes the assumption that the wire lies on the boundary of its convex hull, which I cannot see a good way to justify.

I would be curious to see any improved lower bounds, better solutions than the one given above, or pointers to discussion of this question in the literature.

Solution 1:

A simple lower bound is $\sqrt 5>2.236$. Just choose the $x$-axis along the line joining the endpoints and $y$-axis perpendicular to it. Then, if your wire is $(x(t),y(t)), t\in I$, we must have $\int_I|y'(t)|dt\ge 2$ (you make a swing of width $1$ in the $y$-coordinate but you have to return to the start) and $\int_I|x'(t)|\ge 1$, so $L=\int_I\sqrt{x'(t)^2+y'(t)^2}\ge\sqrt{2^2+1^2}=\sqrt 5$.