Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$).

Definition. The cone $\sigma$ is homogeneous if there are lattice points $a_1,\ldots,a_n\in \sigma \cap \Bbb Z^d$ and a vector $\delta \in \Bbb Z^d$ such that

  1. the monoid $\sigma\cap \Bbb Z^d$ is generated by $a_1,\ldots,a_n$, and
  2. $\delta\cdot a_1 = \cdots = \delta \cdot a_n = 1$,

where $\cdot$ denotes the usual Euclidean inner product.

My Question. Is there a condition on the cone $\sigma$ which guarantees that its dual cone $\sigma^\vee:=\{u\in \Bbb R^d \mid u\cdot a \geq0 \text{ for all } a\in \sigma\}$ is homogeneous?

Note that just being homogeneous is not enough: Let $\sigma = \Bbb R_{\geq0}\{(1,0), (1,1), (1,2), (1,3)\}$, which is homogeneous with $\delta=(1,0)$. Then $\sigma^\vee$ has Hilbert basis $\{(1,0),(0,1),(3,-1)\}$. Since the points $(1,0)$, $(0,1)$, and $(3,-1)$ are non-collinear, $\sigma^\vee$ is not homogeneous.


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