Here:
https://matthewhr.wordpress.com/2014/10/17/heat-ball-and-heat-sphere-mean-value-property/
It's given a reasoning that gives out the heat ball interior and its boundary, separated.
However, I have troubles seeing, why must the boundary be that?
Can one perhaps use the reasoning that since $y$ grows to the positive direction and since the inequality limits $y$ from above, then $y$s at the boundary satisfy the equality part of the inequality?
However, how do I know that $y$ actually attains that value?