AI 中文总结
研究平面中紧致集\(S\subset\mathbb{R}^2\)的平行体积\(V_S(t)\)，引入圆锥可达性条件，通过计算圆锥足部局部管并结合局部施泰纳公式，证明\(V_S\)在\((0,\rho)\)上是次数至多为二的多项式，给出不同形状的相关不变量结果。
AI 中文摘要
对于紧致集\(S\subset\mathbb{R}^2\)，Hug、Last和Weil的局部施泰纳公式通过近端法丛和截断纤维长度\(\min\{t,\delta_S\}\)来表示平行体积\(V_S(t)\)。我们引入圆锥可达性，这是一个有两个要求的几何条件：在有限且分隔良好的奇异集的每个点附近，\(S\)与一个圆锥重合，并且远离这些点的每个近端法纤维长度至少为\(\rho\)。在平面中，这些要求迫使每个链节是圆弧的有限并集，其互补间隙宽度有下界。通过计算每个圆锥的足部局部管并将其与局部施泰纳公式相结合，我们表明\(V_S\)在\((0,\rho)\)上是次数至多为二的多项式，且系数明确；特别地\(\polreach(S)\geq\conreach(S)\)。一维逆命题表明二次壁贡献迫使线性切割函数。具有分段\(C^2\)边界且在其凹角处均匀楔形的紧致域具有正圆锥可达性，并且它们的体积系数由高斯 - 博内型公式给出。对于L形多边形，三个不变量不同：\(\reach(L)=0\)，\(\conreach(L)=\frac{1}{3}\)，\(\polreach(L)=1\)。一个尖点缺口、两个重叠圆盘和线段的康托扇形表明切向接触、凹角处的曲率和退化的链节间隙都会破坏多项式性。
英文摘要
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 $\min\{t,δ_S\}$. 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 $ρ$. 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 $V_S$ is a polynomial of degree at most two on $(0,ρ)$, 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.
