Illumination of complex 1-unconditional convex bodies
We study ordinary and fractional illumination of complex $1$-unconditional convex bodies. In every complex dimension $n$ we give an explicit ordinary bound asymptotic to $3^{n+1}/\sqrt{\pi n}$, and prove $\ill^*(K)\le b_*^n$ and $\ill(K)=O(n\log(n+1)b_*^n)$, where $b_*=2.4805534697\ldots$ is defined by an explicit scalar equation. For normalized non-polydiscs in dimensions two, three and four, the sharp ordinary bounds are $6$, $14$ and $30$. More generally, a full coordinate $(n-1)$-polydisc gives the sharp bound $2^{n+1}-2$ in every dimension $n\ge2$. In dimension two we prove the…
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