arXiv:1805.03557v2 Announce Type: replace Abstract: We establish sharp lower bounds for two geometric functionals which evaluate the global variation of the Gauss map on the boundary of bounded open sets $\Omega\subset\mathbb{R}^d$. These functionals arise as the endpoint case $s=0$ of the Gagliardo seminorm in $W^{s,p}(\partial\Omega)$ of the normal vector field, for $p\in\{1,2\}$. We also show that balls are the unique minimizers. In addition, some applications to the fractional setting $s>0$ are given.
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