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：体积函数V(t)（对于紧集S⊆R^d）定义为到S距离不超过t的点的勒贝格测度。根据几何测度论中的经典结果，在被称为"正触及"的直观且易于解释的充分条件下（该条件可视为凸性概念的推广），体积函数在有限区间内呈现多项式形式。然而，许多不满足正触及条件的简单集合也具有多项式体积函数。据我们所知，目前尚无这类集合的通用简单几何描述。尽管如此，V(t)的多项式特性仍具有重要应用价值，因为多项式系数携带了有用的几何信息——常数项表示S的体积，一阶系数表示（闵可夫斯基意义下的）边界测度。本文聚焦于那些在包含零点的区间上体积函数呈多项式形式（其区间长度称为"多项式触及"）的集合，且该区间长度可能未知。主要目标是通过统计方法，仅利用S内部足够大的随机样本点来近似估计该多项式触及。实际动机在于：当多项式触及值（或其下界）被大致知晓时，可通过多项式逼近的标准方法从样本点估计多项式系数。由此可发展出仅依赖内部样本点而不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。本文探讨了这一思想的理论与实践层面。