  We introduce and begin to explore the mean and median of finite sets of
shapes represented as integral currents. The median can be computed efficiently
in practice, and we focus most of our theoretical and computational attention
on medians. We consider questions on the existence and regularity of medians.
While the median might not exist in all cases, we show that a mass-regularized
median is guaranteed to exist. When the input shapes are modeled by integral
currents with shared boundaries in codimension $1$, we show that the median is
guaranteed to exist, and is contained in the \emph{envelope} of the input
currents. On the other hand, we show that medians can be \emph{wild} in this
setting, and smooth inputs can generate non-smooth medians.
  For higher codimensions, we show that \emph{books} are minimizing for a
finite set of $1$-currents in $\Bbb{R}^3$ with shared boundaries. As part of
this proof, we present a new result in graph theory---that \emph{cozy} graphs
are \emph{comfortable}---which should be of independent interest. Further, we
show that regular points on the median have book-like tangent cones in this
case.
  From the point of view of computation, we study the median shape in the
settings of a finite simplicial complex. When the input shapes are represented
by chains of the simplicial complex, we show that the problem of finding the
median shape can be formulated as an integer linear program. This optimization
problem can be solved as a linear program in practice, thus allowing one to
compute median shapes efficiently.
  We provide open source code implementing our methods, which could also be
used by anyone to experiment with ideas of their own. The software could be
accessed at https://github.com/tbtraltaa/medianshape.
