I just want to mention that the closed form expression of $A_8^*$ as a root of an integer polynomial of degree $42$ is given by Audet, Hansen, Svrtan 2021 in https://link.springer.com/article/10.1007/s10898-020-00908-w The case $n=10$ is contained in https://tinyurl.com/c7hpb752 The approach using Lagrange multipliers is also sketched in https://arxiv.org/abs/2404.01841 , where a representation of the maximum perimeter of a small octagon is derived.
Wow, thanks for pointing all of this out!
Beyond that, the biggest little polyhedra: https://blog.wolfram.com/2015/05/20/biggest-little-polyhedronnew-solutions-in-combinatorial-geometry/
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