Let $\mathbb{S}^d$ be an unit sphere with dimension $d$.
Do we have an integral mean value theorem on the unit sphere? That is we can find a point $t\in\mathbb{S}^d$, such that $$f(t)=\frac{\int_{\mathbb{S}^d}f(y)d\sigma(y)}{\int_{\mathbb{S}^d} d\sigma(y)}?$$
Since we have this result on the interval case: on the interval $[a,b]$, we can find a point $t\in(a,b)$, s.t. $$\int_a^bf(x)dx=f(t)(b-a).$$ Not quite sure this also works on the sphere.
Thank you for any suggestions!
If $f$ is a continuous function $S^d\to \mathbb R$, then the answer is yes, for the same reason as the usual integral mean value theorem (essentially the intermediate value theorem). Namely, the value on the right of your equation is some real number between the maximum and minimum value of $f$. Since $f$ is continuous and $S^d$ is connected (assuming $d>0$), the image of $f$ is an interval and hence contains all numbers between its maximum and minimum.
If $f$ maps to a higher-dimensional space, the answer may be no, for example the inclusion $f: S^1 \to \mathbb R^2$ has average value $0\in \mathbb R^2$ but does not attain that value.
If $f$ is not continuous, then certainly the answer is no in general.