An inequality for the length of isoptic chords of convex bodies

In this note we prove the following result: for every α ∈ ( 0 , π ) and for a given convex body K in the plane, with minimal width w , there exists a chord [ x , y ] with length larger than or equal to w cos α 2 such that there are support lines of K through x and y which form an angle α . Moreover, if there is no such chord with length exceeding w cos α 2 , then K is a Euclidean disc.