The centered maximal operator removes the non-concave Cantor part from the gradient

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $M f(x)>f^*(x)$. We further deduce from this that if the variation measure of $f$ has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of $M f$ can be upgraded to Sobolev regularity.