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?

3 Comments

  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).

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