= Spherical mean
{title2=$M_rf(x)$}
{wiki}
The <spherical mean> of a <function> $f$ on $\mathbb R^n$ averages it over a <sphere> of radius $r>0$ about $x$:
$$
M_rf(x)=\frac1{|S^{n-1}|}\int_{S^{n-1}}f(x+r\omega)\,d\omega.
$$
Here $d\omega$ is surface measure on the unit <sphere> and $|S^{n-1}|$ is its total area. Using the unit <sphere> absorbs the physical surface-area factor $r^{n-1}$ into the change of variables. The expression requires an integrable restriction to that sphere; continuity near it is one sufficient condition.
For $f(y)=|y|^2$, expanding $|x+r\omega|^2$ and using the zero average of $\omega$ gives $M_rf(x)=|x|^2+r^2$. By contrast, the <mean value property for harmonic functions> gives $M_ru(x)=u(x)$ when $u$ is <harmonic> on a neighborhood of the closed <ball>. Averaging over a boundary <sphere> and averaging over the whole <ball> are different operations, even though both have a mean-value property for <harmonic functions>.
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