2026/07/27 by Alejandro Cholaquidis
#math.GT
For a compact set S⊂\R2, the local Steiner formula of Hug, Last and Weil expresses the parallel volume VS(t) through the proximal normal bundle and the truncated fiber lengths min\t,δS\. We introduce conic reach, a geometric condition with two requirements: S coincides with a cone near each point of a finite, well-separated singular set, and every proximal normal fiber away from those points has length at least ρ. In the plane, these requirements force every link to be a finite union of circular arcs whose complementary gaps have width bounded below. Computing the feet-localized tube of each cone and combining it with the local Steiner formula, we show that VS is a polynomial of degree at most two on (0,ρ), with explicit coefficients; in particular \polreach(S)≥\conreach(S). A one-dimensional converse shows that a quadratic wall contribution forces a linear cut function. Compact domains with piecewise-C2 boundary, uniformly wedge-like at their reentrant corners, have positive conic reach, and their volume coefficients are given by a Gauss--Bonnet-type formula. For the L-shaped polygon the three invariants separate: \reach(L)=0, \conreach(L)=\tfrac13, \polreach(L)=1. A cuspidal notch, two overlapping discs and a Cantor fan of segments show that tangential contact, curvature at a reentrant corner and degenerating link gaps each destroy polynomiality.