Given upper bound of the width, which closed convex curve has the largest area?

Question

The "width" represents the distance between two parallel supporting lines.

I guess it must be the constant width curves, among which the circle has the maximum area. But I cannot prove this.

Answer 1

The maximum width of a bounded planar set $S$ is just equal to its diameter $d$ (the maximum distance between any two points), because for any two parallel lines touching $S$, the distance between their points of contact is at least the width (and hence the width is $\le d$), and by taking the lines to be perpendicular to a diameter of $S$ we can attain this bound.

So your question amounts to "Given an upper bound on the diameter, which convex curve has the largest area?". For this, we can use the isodiametric inequality: $4A\le \pi d^2$, or equivalently, the disc has the largest volume among sets of a fixed diameter in the plane. So your guess is correct.

This PDF goes into more detail about a proof of this fact.

(In fact, everything in this answer holds for arbitrary sets (no convexity necessary) in $\mathbb{R}^n$, if we replace "parallel lines" with "parallel hyperplanes of dimension $n-1$" and "disc" with "$n$-dimensional ball".)