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    $\begingroup$ To integrate a function $f(x,y,z,w)$ over the unit 4-sphere's surface you can integrate it in explicit hyperspherical coordinates: NIntegrate[f[Cos[θ], Cos[χ]*Sin[θ], Cos[φ]*Sin[θ]*Sin[χ], Sin[θ]*Sin[φ]*Sin[χ]] * Sin[θ]^2*Sin[χ], {θ, 0, π}, {χ, 0, π}, {φ, 0, 2 π}]. $\endgroup$ Commented Aug 21, 2021 at 16:11
  • $\begingroup$ Thanks! That might help... $\endgroup$ Commented Aug 21, 2021 at 16:53
  • $\begingroup$ Thank you, that indeed works and I can accept that as an aswer. Do you perhaps know how to run NMaximize now with this? It seems that NMaximize[improvement[a[[1 ;; d]], a[[d + 1 ;; 2*d]], a[[2*d + 1 ;; 3*d]], a[[3*d + 1 ;; 4*d]]], a [Element] Sphere[4*d]] goes through only one iteration and returns NMaximize::ivar: True is not a valid variable., and then NMaximize[4.90928724127370, True]. Improvement is the sum of term functions as above for $k = 1, 2, 3, 4$. Thank you again! $\endgroup$ Commented Aug 22, 2021 at 10:42
  • $\begingroup$ Please ask a separate question for this follow-up! $\endgroup$ Commented Aug 22, 2021 at 15:18