Conic reach and polynomial parallel volume in the plane

Abstract

For a compact set $S\subset\R^2$, the local Steiner formula of Hug, Last and Weil expresses the parallel volume $V_S(t)$ through the proximal normal bundle and the truncated fiber lengths $\min\{t,\delta_S\}$. 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 $\rho$. 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 $V_S$ is a polynomial of degree at most two on $(0,\rho)$, with explicit coefficients; in particular $\polreach(S)\ge\conreach(S)$. A one-dimensional converse shows that a quadratic wall contribution forces a linear cut function. Compact domains with piecewise-$C^2$ 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.