  We introduce and discuss the concept of \emph{$n$-distance}, a generalization
to $n$ elements of the classical notion of distance obtained by replacing the
triangle inequality with the so-called simplex inequality $$ d(x_1, \ldots,
x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots,
x_n, z \in X, $$ where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the
function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide
several examples of $n$-distances, and for each of them we investigate the
infimum of the set of real numbers $K\in\left]0,1\right]$ for which the
inequality above holds. We also introduce a generalization of the concept of
$n$-distance obtained by replacing in the simplex inequality the sum function
with an arbitrary symmetric function.
