Pompeiu's theorem

Pompeiu's theorem is a geometric result concerning the relationships between geometric shapes and their properties. Specifically, it states that if \( S \) is a bounded measurable set in the Euclidean space \( \mathbb{R}^n \), and if \( f: \mathbb{R}^n \to \mathbb{R} \) is a continuous function such that the integral of \( f \) over \( S \) is zero (i.e.