  Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with
$L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if
every chord of $K$ tangent to $L$ has length equal to a given fixed value
$\lambda$. J. Barker and D. Larman proved that if $L$ is a ball, then $K$ is a
ball concentric with $L$. In this paper we prove that there exist an infinite
number of closed curves, different from circles, which possess an equichordal
convex body. If the dimension of the space is more than or equal to 3, then
only Euclidean balls possess an equichordal convex body. We also prove some
results about isoptic curves and give relations between isoptic curves and
convex rotors in the plane.
