The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V(t)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.

翻译：紧集S⊆R^d的体积函数V(t)定义为距S距离不超过t的点集的勒贝格测度。根据几何测度论中的经典结论，在直观且易于解释的充分条件（称为“正触及”）下——该条件可视为凸性的延伸——体积函数至少在有限区间内具有多项式形式。然而，许多不满足正触及条件的简单集合同样具有多项式形式的体积函数。据我们所知，目前尚缺乏对这类集合通用且简洁的几何描述。但V(t)的多项式特性具有重要应用价值，因其多项式系数承载着若干有用的几何信息：常数项对应S的体积，一阶系数对应（闵可夫斯基意义下的）边界测度。本文聚焦于体积函数在包含零点的某区间（其长度称为“多项式触及”）上呈多项式形式的集合，且该区间长度可能未知。我们的主要目标是仅利用S内部足够大的随机样本点，通过统计方法逼近该多项式触及。实际动机十分简单：当多项式触及值（或其下界）近似已知时，可通过标准多项式逼近方法从样本点估计多项式系数。由此获得一种仅依赖内部样本点、无需使用任何平滑参数的通用方法，用于估计集合的体积和边界测度。本文系统探讨了这一思路的理论与实践层面。