Skip to main content
MathOverflow

Return to Question

edited title; deleted 20 characters in body
Source Link
Camilo Sarmiento
Camilo Sarmiento
  • 902
  • 9
  • 30

Which fans are secondary? (or which polytopespolytopal) fans/polytopes are secondary?)

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.

The secondary fan of $P$ is a complete polyhedralpolytopal fan in $\mathbb{R}^n$ with $d$-dimensional linearity space whose cones are are in one-to-one correspondence with regular subdivisions of $P$ (see the book by GKZ for a detailed description).

Is there any characterization of those complete polyhedralpolytopal fans that are the secondary fan of some polytope?

I know that (the pointed part of) secondary fans can be described as the mutual refinement of all simplicial cones spanned by elements in the Gale transform of some polytope. However, I don't know how to get such a description if all I have is the inequalities defining the cones in thea polytopal fan (let alone what could go wrong when trying to do that.)

Which fans are secondary? (or which polytopes are secondary?)

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.

The secondary fan of $P$ is a complete polyhedral fan in $\mathbb{R}^n$ with $d$-dimensional linearity space whose cones are are in one-to-one correspondence with regular subdivisions of $P$ (see the book by GKZ for a detailed description).

Is there any characterization of those complete polyhedral fans that are the secondary fan of some polytope?

I know that (the pointed part of) secondary fans can be described as the mutual refinement of all simplicial cones spanned by elements in the Gale transform of some polytope. However, I don't know how to get such a description if all I have is the inequalities defining the cones in the fan (let alone what could go wrong when trying to do that.)

Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.

The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional linearity space whose cones are are in one-to-one correspondence with regular subdivisions of $P$ (see the book by GKZ for a detailed description).

Is there any characterization of those polytopal fans that are the secondary fan of some polytope?

I know that (the pointed part of) secondary fans can be described as the mutual refinement of all simplicial cones spanned by elements in the Gale transform of some polytope. However, I don't know how to get such a description if all I have is the inequalities defining the cones in a polytopal fan (let alone what could go wrong when trying to do that.)

Source Link
Camilo Sarmiento
Camilo Sarmiento
  • 902
  • 9
  • 30

Which fans are secondary? (or which polytopes are secondary?)

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.

The secondary fan of $P$ is a complete polyhedral fan in $\mathbb{R}^n$ with $d$-dimensional linearity space whose cones are are in one-to-one correspondence with regular subdivisions of $P$ (see the book by GKZ for a detailed description).

Is there any characterization of those complete polyhedral fans that are the secondary fan of some polytope?

I know that (the pointed part of) secondary fans can be described as the mutual refinement of all simplicial cones spanned by elements in the Gale transform of some polytope. However, I don't know how to get such a description if all I have is the inequalities defining the cones in the fan (let alone what could go wrong when trying to do that.)