Some results about equichordal convex bodies

Abstract

Let $K$ and $L$ be two convex bodies in ${\mathbb{R}}^n$, $n\ge 2$, 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.