Conical $3$-uniform measures: A family of new examples and characterizations

Abstract

28A33, 49Q15. Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform measures. Indeed, the question of whether there exist any non-flat uniform measures beside the one supported on the Kowalski–Preiss cone has been open for 30 years, ever since Kowalski and Preiss classified $n$‑uniform measures in $\mathbb{R}^{n+1}$. In this paper, we answer the question and construct an infinite family of $3$‑uniform measures in arbitrary codimension. We define a notion of distance symmetry for points and prove that every collection of $2$‑spheres whose centers are distance symmetric gives rise to a $3$‑uniform measure. We then develop a combinatorial method to systematically produce distance symmetric points. We also classify conical $3$‑uniform measures in $\mathbb{R}^5$ by proving that they all arise from distance symmetric spheres.