BV Regularity and Differentiability Properties of a Class of Upper Semicontinuous Functions

We study a class of upper semicontinuous functions $f:\mathbb R^d\to\mathbb R$ whose hypograph $\mathrm{hypo}\,f$ satisfies a geometric regularity property, namely: there exist $c>0$, $\theta\in]0,1]$ such that for each $P$ on the boundary of $\mathrm{hypo}\,f$ there exists a unitary Fr\'echet (outer) normal $v\in N^F_{\mathrm{hypo}\,f}(P)\cap\mathbb S^d$ to $\mathrm{hypo}\,f$ with \[\langle v,P-Q\rangle\le c \|P-Q\|^{1+\theta}\qquad\textrm{for every }Q\in\mathrm{hypo}\,f.\