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  "body": "\n## Summary\n\nThe spherical mean transform sends a function $f$ on $\\R^n$ to a function $M f(x, r)$.\nIts value is the average of $f$ over the [[sphere|P9CM4GQ2T0]] of center $x$ and radius $r$.\nA continuous function is [[harmonic|0DEB0JMN0W]] exactly when all its spherical means equal its value at the center.\nFor a function of class $C^2$ the means satisfy the Darboux equation.\nThis reduces the [[wave equation|5EHD2XBQDG]] in $\\R^n$ to an equation in one space variable.\nRecovering $f$ from its means over spheres with centers on a fixed surface is the mathematical core of thermoacoustic and photoacoustic tomography.\n\n## Explanations\n\n### L1 human\n\nStand at a point $x$, draw a sphere of radius $r$ around it, and average the values of $f$ on that sphere.\nDoing this for every center and every radius gives the spherical mean transform.\nFor example, for $f(x) = \\abs{x}^2$ the mean over the sphere of center $x$ and radius $r$ is $\\abs{x}^2 + r^2$.\nFor radius $0$ the mean is just the value $f(x)$, and for a harmonic function it stays the same for all radii.\n",
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