  It is known that the distribution of an integrable random vector $\xi$ in
$\mathbb{R}^d$ is uniquely determined by a $(d+1)$-dimensional convex body
called the lift zonoid of $\xi$. This concept is generalised to define the lift
expectation of random convex bodies. However, the unique identification
property of distributions is lost; it is shown that the lift expectation
uniquely identifies only one-dimensional distributions of the support function,
and so different random convex bodies may share the same lift expectation. The
extent of this nonuniqueness is analysed and it is related to the
identification of random convex functions using only their one-dimensional
marginals. Applications to construction of depth-trimmed regions and partial
ordering of random convex bodies are also mentioned.
