# Some more geometry in the secondary fan

## Frank Sottile, Texas A&M University, College Station

## January 6, 2015

A polyhedral subdivision of finitely many points $A$ in $\mathbb{R}^d$
is a decomposition of its convex hull into non-overlapping
polytopes with vertices in $A$. It is regular if the polytopes
are projections of facets of a polyhedron in $\mathbb{R}^{d+1}$. Gelfand,
Kapranov, and Zelevinsky introduced the secondary fan of $A$,
which encodes all regular subdivisions of $A$.
I will talk on work with Postinghel and Villamizar that
explains another geometric interpretation of the secondary fan.
The points $A$ give a curvilinear copy of the convex hull of
A in the probability simplex whose vertices correspond to $A$
which we call the real toric variety of $A$. We consider
possible Hausdorff limits of translations of this real toric
variety. We show that the possible limits correspond to the
cones of the secondary fan. Our main tool is a study of
sequences in the ambient space of the secondary fan.