אלון שכטר

math.MG — Metric Geometry

Illumination of complex convex bodies

Contributed by אלון שכטר

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 sharp bound $14$ for every normalized complex $1$-unconditional body in $\mathbb{C}^3$ other than $\mathbb{D}^3$, and the sharp bound $30$ in $\mathbb{C}^4$ for every normalized non-polydisc. The four-dimensional proof includes one finite covering assertion verified by exact integer arithmetic; its complete verifier is included. Under full coordinate permutation symmetry, we improve the three-dimensional bound to $12$ by an explicit twelve-vector construction and an exact finite covering certificate. For complex Hanner bodies, we compute both illumination numbers exactly and determine the equality cases in their dimension bounds. If every extreme point has all coordinates nonzero, we prove $\operatorname{ill}(K)\le2^{n+1}-1$ and a quantitative strict fractional bound $\operatorname{ill}^*(K)<2^n$ for normalized non-polydiscs. For real $1$-symmetric non-cubes we obtain bounds $12$ in dimension four, $22$ in dimension five, and $5\cdot2^{n-3}$ for every $n\ge6$. We also determine the leading asymptotic maximum in the real $1$-symmetric non-cube class: it is $(1+o(1))2^{n-1}$. An explicit binary construction proves this without an external coding theorem. A connection to connected domination in cube graphs improves the upper estimate to $2^{n-1}(1+(1+o(1))/n)$.