https://www.cut-the-knot.org/do_you_know/isoperimetric.shtml

or

https://mathdl.maa.org/images/upload_library/22/Ford/blasjo526.pdf

However, the 3d case is much more involved. Intuitively, for the optimum body (if such exists), any plane that halves the area must halves the volume. The hard part is to prove from here that, for any plane, there is a parallel plane in which the body is symmetric. From here, one gets central symmetry as well. Summing up, sphere is the only possibility to support an extreme V²/S³ (Volume and Surface area).