A Fine Property of Sets at Points of Lebesgue Density

Abstract

Consider an integer n≥2, m∈[n,+∞) and put α‾:=m−1n−1. Moreover: ∙ Let γx:(0,+∞)→Rn be the line through x=(x1,…,xn)∈Rn defined as γx(t):=(tx1,tα‾x2,…,tα‾xn); ∙ If T is any motion of Rn, then let γx(T) be the line through x=(x1,…,xn)∈Rn defined as γx(T):=T∘γT−1(x); ∙ Let H1 denote the one-dimensional Hausdorff measure in Rn. The main goal of this paper is to prove the following property: If x0 is an m-density point of a Lebesgue measurable set E and T is an arbitrary motion of Rn mapping the origin to x0, then we have lim supt→0+ H1(E∩γx(T)((0,t])) H1(γx(T)((0,t]))=1 for almost every x∈T({1}×Rn−1). An application of this result to locally finite perimeter sets is provided.