Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Let $(Y,\|\cdot\|)$ be a Banach space. Given a Bochner integrable map $f:X\to Y$ with $\|f\| \in L^2(X,\mu)$. The mean value of $f$ over $X$, denoted by $f_X$, is defined by $$\frac{1}{\mu(X)}\int_{X}fd\mu.$$

Question: Is it true that $f_X$ is a minimum of the function $$Y\ni y \mapsto \int_X \|f(x)-y\|^2 d\mu(x) ?$$