Measures of $N$-fold symmetry for convex sets

If a convex set $S$ is $3$-fold symmetric about a point $0 \in S$, then any $3$-star contained in $S$ with vertex 0 is no smaller than any other parallel $3$-star contained in $S$. In this paper, among other results, we establish the converse. Consequently, we find two measures of $n$-fold symmetry, one for $n = 2,3$ and the other for each $n \geqq 2$.