A BV function is minimally singular iff its singular vertical distance to some point vanishes almost everywhere, and over a wide class of open connected domains this property is equivalent to rigidity for Steiner's inequality.
Cagnetti , Rigidity for perimeter inequalities under symmetrization: state of the art and open problems , Port
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Minimally singular functions and the rigidity problem for Steiner's perimeter inequality
A BV function is minimally singular iff its singular vertical distance to some point vanishes almost everywhere, and over a wide class of open connected domains this property is equivalent to rigidity for Steiner's inequality.