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

## TL;DR

This paper establishes sharp lower bounds on the measure of unions of essentially distinct $ ext{delta}$-tubes in $ ext{R}^n$, characterizes extremal configurations, and introduces a new convexity measurement via the X-ray transform.

## Contribution

It provides the first asymptotically sharp bounds for unions of essentially distinct $ ext{delta}$-tubes and introduces a novel convexity measurement based on the X-ray transform.

## Key findings

- Sharp lower bounds for union measure of $ ext{delta}$-tubes
- Characterization of extremal configurations
- New convexity measurement method

## Abstract

We say two $\delta$-tubes (dimension $\delta\times\cdots\times\delta\times1$) 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.