In a comment Mark Meckes asked about an argument to reduce the $n\geq 4$ case to the $n=3$ case. Let $n\geq n'\geq 3.$ A smooth function on a convex set $B$ is constant if and only if the restriction to every $B\cap S,$ where $S$ is an $(n'-2)$-dimensional affine subspace, is constant. At least intuititvely, this means the pushforward measure on $B^{n-2}$ from the projection $S^{n-1}\to B^{n-2}$ is uniform if and only if there is a uniform conditional distribution for each $(n'-2)$-dimensional subspace. But fixing such a subspace corresponds to restricting to a scaled version of $S^{n'-1}\to B^{n'-2}.$ This implies that the results for $n$ and $n'$ are equivalent.

So if you accept Archimedes' theorem, you get the result for all $n.$ Alternatively you might find it more intuitive to prove the result for $n=4.$ This calculation is not too bad. ["On the volumes of balls" by Blass and Schaunuel][1] uses the parameterization $s_i=\tfrac 1 2 r_i^2$ to give a two-to-one map $B^2\times B^2\to B^4$ preserving the volume form $ds_1d\theta_1ds_2d\theta_2.$ Specifically, $(s_1,\theta_1,s_2,\theta_2)\to (s_1-s_2,\theta_1,s_2,\theta_2)$ for $s_1>s_2.$ For $s_1=1$ this defines a volume preserving map $S^1\times B^2\to S^3.$


  [1]: http://www.math.lsa.umich.edu/~ablass/geom.html