If $X$ is a connected convex set in $S^{n-1}(1)$, then what is $\partial X$ ?
Recall the following definition.
Definition : $X$ is a convex subset of $S^{n-1}(1)$ if any two points in $X$ can be joined by distance minimizing geodesic which lies in $X$
Except finite points, is it a union of $C^1$-paths ? And $\partial X$ has a finite variation ? : Let $x_i^n \in \partial X$ with $i=1, \cdots, n$ such that $|x_i^n-x_{i+1}^n|=|x_1^n-x_n^n|$ for all $1\leq i \leq n-1$ where $|\cdot |$ is a distance function on $S^{n-1}(1)$ Then $lim_{n\rightarrow \infty} n|x_1-x_2| < \infty$.