Let $\mathcal{S}^{n-1}$ denote the $(n-1)$-dimensional hypersphere, and let $\sigma$ denote the normalized Hausdorff measure on $\mathcal{S}^{n-1}$ (so $\sigma(\mathcal{S}^{n-1}) = 1$).
One poses the question: Let $a > 0$ be fixed. Among all centrally-symmetric subsets $A \subseteq \mathcal{S}^{n-1}$ satisfying $\sigma(A) = a$, what is the $A$ which minimizes the perimeter $\mathrm{vol}_{n-2}(\partial A)$?
If one drops the requirement that $A$ is centrally-symmetric, we have the classical isoperimetric inequality on the sphere, for which the minimizer is known (spherical cap).
Reimposing the requirement that $A$ is centrally-symmetric, experimentation in the case $n=3$ suggests the following phenomenon: there exists a threshold $\eta$ such that if $a<\eta$, the minimizer looks like two antipodal spherical caps, and if $a > \eta$, the minimizer looks like a strip.
What is known about this problem? I couldn't find a solution anywhere.
According to the paper "Isoperimetry and volume preserving stability in real projective spaces" by Celso Viana posted on the arXiv in 2019, the solution to the isoperimetric problem in round real projective space $\mathbb{RP}^n$ consists of tubular neighborhoods of some $k$-dimensional subspace $T \subseteq \mathbb{RP}^n$, where the optimal value of $k$ will depend on the prescribed volume of the subset $A$ in the question above.
This answers the original problem because a centrally-symmetric subset $A \subseteq \mathcal{S}^{n-1}$ can be viewed as a subset of projective space $\mathbb{RP}^{n-1}$. Where $\mathbb{RP}^{n-1}$ is given the standard metric.
Link: https://arxiv.org/abs/1907.09445