Illumination of complex convex bodies
For every complex $1$-unconditional convex body $K\subset\mathbb{C}^n$, we prove \[ \operatorname{ill}(K)\le (2^{m+1}-1)\binom nm+\sum_{k=m+1}^{n}\binom nk-(n-m), \qquad m=\lceil2n/3\rceil. \] This bound is asymptotic to $3^{n+1}/\sqrt{\pi n}$. We further obtain $\operatorname{ill}^*(K)\le b_*^n$ and $\operatorname{ill}(K)=O(n\log(n+1)b_*^n)$, where $b_*=2.4805534697\ldots$ is defined by an explicit scalar equation. For normalized bodies other than $\mathbb{D}^n$ that contain a full coordinate $(n-1)$-polydisc, we prove the sharp bound $\operatorname{ill}(K)\le2^{n+1}-2$. We also prove the…
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