On the union of essentially distinct $\delta$-tubes

Abstract

We say two $\delta$-tubes (dimension $\delta \times \cdots \times \delta \times 1$) in ${\mathbb{R}}^n$ are essentially distinct if the measure of their intersection is smaller than a half of a single $\delta$-tube. For a collection of essentially distinct $\delta$-tubes, we give the asymptotically sharp lower bound for the measure of their union. Then we characterize all sharp examples. We will give a new measurement of convexity based on the X-ray transform.