K-Polystable Toric Fano Varieties With Small Alpha Invariants

For every $n\ge2$ the paper exhibits an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety whose alpha invariant is exactly $\tfrac{2}{2n+1}$, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between $\tfrac{1}{n+1}$ and $\tfrac1n$.