On the convexity of $N$-Chebyshev sets

We define $N$-Chebyshev sets in a Banach space $X$ for every positive integer $N$ (when $N=1$, these are ordinary Chebyshev sets) and study conditions that guarantee their convexity. In particular, we prove that all $N$-Chebyshev sets are convex when $N$ is even and $X$ is uniformly convex or $N\geqslant 3$ is odd and $X$ is smooth uniformly convex.