Universal volume bounds for reflexive toric Calabi–Yau cones

Let $X(\Delta_{n-1})$ be the toric variety associated with a reflexive toric diagram $\Delta_{n-1}$, and let $\widetilde{X(\Delta_{n-1})}$ denote the relevant resolution. Let $V_{\min}$ be the minimum volume of the corresponding Sasaki–Einstein $(2n-1)$-manifold, and let $m_n$ satisfy $m_n>m_{n+1}$. Universal volume-bound conjecture. For every $n$,

$$\frac{1}{\chi(\widetilde{X(\Delta_{n-1})})}\leq V_{\min}\leq m_n\int c_1^{n-1}(\widetilde{X(\Delta_{n-1})}).$$

The bounds are expressed through the Euler number and the first Chern number of the associated toric variety. The paper attributes this proposed universal extension to previously studied cases $n=3$ and $4$; its validity for arbitrary $n$ remains open.