## Connecting two points in the range of a vector measure

### Volume 153 / 2018

#### Abstract

Let $\mu =( \mu _{1},\ldots ,\mu _{n}) $ be a nonatomic vector measure defined on measurable subsets $ \cal {B} $ of the unit interval $I= [0,1] $. Denote by $ {\cal U}(k) $ the collection of all sets that are unions of at most $ k $ disjoint subintervals of $ I $. We show that if $ A \in {\cal U}(k) $ then the line segment connecting $0\in \mathbb {R}^n$ and $ \mu (A) $ is contained in $ \mu ( {\cal U}(n+k-1))$. Moreover, if $B,C \in {\cal U}(k) $ then the line segment connecting $\mu (B)$ and $ \mu (C)$ is a subset of $ \mu ( {\cal U}(2n+4k-3))$. This result is used to give yet another proof of the Lyapunov convexity theorem. We also discuss the two-dimensional case for specific vector measures.