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1. In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius $\frac12$. His elegant, unpublished argument is reproduced here with his kind permission.
THEOREM 1 (A. Meir). Every plane arc of length $L$ lies in some closed semidisk of radius $L / 2$.
Proof. The assertion is clear for closed curves, for such a curve plainly lies in a semidisk of radius $L / 2$ centered at a point of contact of any support line of the curve. Let $\Gamma$ be an arc of length $L$ having distinct endpoints $P$ and $Q$, let $l$ be a line of support parallel to the line $P Q$ and touching $\Gamma$ at a point $R$, and let $P^{\prime}$ and $Q^{\prime}$ be the points symmetric to $P$ and $Q$ in $l$ (Figure 1). Let $O$ be the point in which the lines $P Q^{\prime}$ and $Q P^{\prime}$ meet $l$. Each point $X$ on $\Gamma$ lies between $R$ and $P$ or between $R$ and $Q$ along $\Gamma$, and we may suppose that $X$ lies between $R$ and $P$. Because the median of a triangle is shorter than the average of the lengths of the two adjacent sides,
$$
O X \leq \frac{1}{2}\left(X P+X Q^{\prime}\right) \leq \frac{1}{2}(P X+X R+R Q) \leq \frac{1}{2} L
$$
Thus $\Gamma$ lies in the semidisk of radius $L / 2$ and edge $l$ centered at the point $O$.
In §3 of this note we generalize this result to circular sectors and show that there is a sector of area less than $0.3451 L^{2}$ that can accommodate every arc of length $L$. Meir's semidisk has area $\pi L^{2} / 8 \approx 0.3927 L^{2}$.
Section 2 is devoted to a characterization of circular sectors that contain a translate of every closed curve of length $L$.
In §4 we show that the least area of a convex set that contains a translate of every closed curve of length $L$ lies between $0.15544 L^{2}$ and $0.15900 L^{2}$, and in §5 we show that the least area of a convex set that contains a displacement of every arc of length $L$ lies between $0.21946 L^{2}$ and $0.34423 L^{2}$.
2. A circular sector is circumscribed about a curve if the curve lies in the sector and has a point on the circular boundary arc and a point on each of the boundary radii. We begin with a result about circular sectors that are circumscribed about a closed curve of length $L$.
Let $\operatorname{Csc}x=\csc x$ when $0<x<\pi / 2$ and $\operatorname{Csc}x=1$ when $\pi / 2 \leq x \leq \pi$. For $r>0$ and $0<\theta \leq \pi$, we denote the circular sector with radius $r$ and vertex angle $\theta$ by $S(r, \theta)$.
LEMMA 2. If a circular sector $S(r, \theta)$ is circumscribed about a closed curve of length $L$, then $r \leq(L / 2) \operatorname{Csc} \theta$.
Proof. Let the sector $S(r, \theta)=\langle\mathrm{BAC}\rangle$ be circumscribed about a closed curve $\Gamma$ of length $L$, and let $X, Y$, and $Z$ be points of $\Gamma$ on the circular $\operatorname{arc} B C$ and radial segments $A B$ and $A C$, respectively. The perimeter $p$ of $\triangle X Y Z$ is at most $L$, and $p$ equals $L$ precisely when the curve $\Gamma$ coincides with $\triangle X Y Z$. Let $X^{\prime}$ and $X^{\prime \prime}$ be the points symmetric to $X$ in the lines $A B$ and $A C$ respectively. If $\theta<\pi / 2$, then
$$
p=X^{\prime} Y+Y Z+Z X^{\prime \prime} \geq X^{\prime} X^{\prime \prime}=2 r \sin \theta .
$$
If $\pi / 2 \leq \theta \leq \pi$, then
$$
p=X^{\prime} Y+Y Z+Z X^{\prime \prime} \geq X^{\prime} Z+Z X^{\prime \prime} \geq X^{\prime} A+A X^{\prime \prime}=2 r .
$$
In either case,
$$
r \leq \frac{1}{2} p \operatorname{Csc} \theta \leq \frac{1}{2} L \operatorname{Csc} \theta .
$$
When $\theta$ is acute, the equality occurs precisely when $\Gamma$ coincides with $\triangle X Y Z$ and the points $X^{\prime}, Z, Y$, and $X^{\prime \prime}$ are collinear. When $\theta$ is not acute, the equality occurs precisely when $\Gamma$ is a radial segment (traversed twice).
A compact, convex set in the plane is a translation cover for a family of plane arcs if for each arc in the family there is a translation that carries the arc into the set. We can use Lemma 2 to characterize sectorial translation covers for the family
Wetzel, J. (1973). Sectorial Covers for Curves of Constant Length. Canadian Mathematical Bulletin, 16(3), 367-375.
sectorial-covers-for-curves-of-constant-length.pdf
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Schaer, J., Wetzel, J.E. Boxes for curves of constant length. Israel J. Math. 12, 257–265 (1972).
Boxes for curves of constant length.pdf
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