Definition Simplex [lcc-002S]

The free convex space on a finite set \{0, \dots n\} of n+1 elements is called the n-simplex and denoted \Delta ^n (the reason for the apparent mismatch of numbering is that the n-simplex is n-dimensional). Note that an element of \Delta ^n is a tuple (s_i)_{i=0,\dots , n} so that \sum _i s_i = 1 and s_i \geq 0. In particular, \Delta ^1 \cong [0,1].