## Secondary Polytopes

An ingenious construction of Gelfand, Kapranov, and
Zelevinsky [1] associates to each triangulation of a set of n
points *x*_{1},...,x_{n} in R^{d} a
point in R^{n}. The *i*th coordinate of this new
point is the total volume of all simplices incident to
x_{i} in the triangulation. The convex hull of these
new points is called the *secondary polytope * of the point
configuration. It has dimension *n-d-1*, and its vertices
correspond to the so-called *regular triangulations* of the
point set.

The *flip graph* of a point configuration is the graph
whose vertices are the triangulations of the configuration, and
where two vertices are connected by an edge if there exists a
bistellar operation transforming one triangulation into the
other. The 1-skeleton of the secondary polytope is therefore the
subgraph of the flip graph induced by the regular triangulations
of the point configuration.

It is important to realize that the secondary polytope and
the flip graph are **NOT** combinatorial invariants of a
given point configuration, but depend on its metrical
properties.

#### Reference

- I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky:
*Discriminants, Resultants, and Multidimensional
Determinants*, Birkhäuser (1994).