## Should start groping for 5-dim’l cousin of V_22

$V_{22}$ can be characterized  as the smooth Fano of index 1 with minimal cohomology. One way to construct its D3 is by assigning 1’s to the vertices of a terminal self-dual polytope.

Are there terminal self-dual polytopes in dim 5 that produce D5’s whose matrix entries are positive integers?

Am I right in believing that one can search for self-duals considerably more easily than for arbitrary reflexives?

1. Sergei says:

A. If we look for terminal self-dual polytope then in dimensions up to 3 it is unique

d=1. interval – $P^1$,
d=2. honeycomb – $S_6$ (smooth del Pezzo of degree 6)
d=3. $T_{22}$ – STD of $V_{22}$

In dimension 4 there is a unique terminal self-dual 24-cell,
and Benjamin Nill told me once some uniqueness result,
but I forgot the correct formulation – it could be there is nothing except the unique 24-cell.

In dimension 8 there is a polytope corresponding to E_8 root system (convex hull of roots).

B. $P^3, Q, V_5$ are not self-dual as lattice polytopes, however their duals are their homotheties.

Property “terminal self-dual” can be weakened in two directions to produce two other interesting properties:

I. When dual of terminal polytope is terminal.
II. When Gorenstein polytope is combinatorially self-dual.
II+. When polytope and its dual are homothety-equivalent.

2. Vasily says:

I wonder if there might be a Monte Carlo method to arrive at self-duals by iteration. Start with something random, generate a dual, adjust it to become a lattice polytope, generate a dual, continue.

3. Sergey says:

Then you just end up with reflexive polytope.
Or you need to adjust somehow to terminal.
And then you end with pair of dual reflexive terminals (problem of type I above).