  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.
