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Illumination of complex convex bodies

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometry

Contributed by אלון שכטר ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

For every complex 11-unconditional convex body K⊂CnK\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 3n+1/πn3^{n+1}/\sqrt{\pi n}. We further obtain ill⁡∗(K)≤b∗n\operatorname{ill}^*(K)\le b_*^n and ill⁡(K)=O(nlog⁡(n+1)b∗n)\operatorname{ill}(K)=O(n\log(n+1)b_*^n), where b∗=2.4805534697…b_*=2.4805534697\ldots is defined by an explicit scalar equation. For normalized bodies other than Dn\mathbb{D}^n that contain a full coordinate (n−1)(n-1)-polydisc, we prove the sharp bound ill⁡(K)≤2n+1−2\operatorname{ill}(K)\le2^{n+1}-2. We also prove the sharp bound 1414 for every normalized complex 11-unconditional body in C3\mathbb{C}^3 other than D3\mathbb{D}^3, and the sharp bound 3030 in C4\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 1212 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 ill⁡(K)≤2n+1−1\operatorname{ill}(K)\le2^{n+1}-1 and a quantitative strict fractional bound ill⁡∗(K)<2n\operatorname{ill}^*(K)<2^n for normalized non-polydiscs. For real 11-symmetric non-cubes we obtain bounds 1212 in dimension four, 2222 in dimension five, and 5⋅2n−35\cdot2^{n-3} for every n≥6n\ge6. We also determine the leading asymptotic maximum in the real 11-symmetric non-cube class: it is (1+o(1))2n−1(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 2n−1(1+(1+o(1))/n)2^{n-1}(1+(1+o(1))/n).

Provenance statement

Results generated with Chatgpt Astra-6. The bounds for 1-unconditional have been checked by a human.

References

  1. L. Rotem, A. Schejter, and B. A. Slomka, The complex Illumination problem , Combinatorica 46 , article 3 (2026). doi:10.1007/s00493-025-00195-7 (https://doi.org/10.1007/s00493-025-00195-7). For the exact polydisc theorem and the explicit cyclic construction, see Theorem 1.4 and the proof of Proposition 2.7 in arXiv:2410.12021v1 (https://arxiv.org/abs/2410.12021v1) (2024).arXivDOI
  2. L. Rotem, A. Schejter and B. Slomka, Illumination of Real and Complex 1-Symmetric Convex Bodies , manuscript, October 2026, Theorems 1.4--1.5, Propositions 2.8--2.9 and 3.1, Lemma 4.3 and Proposition 4.4.
  3. W. R. Sun and B.-H. Vritsiou, Illuminating 11-unconditional convex bodies in R3\mathbb R^3 and R4\mathbb R^4, and certain cases in higher dimensions , arXiv:2407.11331v1 (https://arxiv.org/abs/2407.11331v1) (2024), Theorem 5 and Lemmas 33 and 35.arXiv
  4. J. R. Griggs, Spanning Trees and Domination in Hypercubes , Integers 21A (2021), Paper A13, 11 pp., Theorems 2 and 4. Full text (https://math.colgate.edu/~integers/graham13/graham13.pdf).link

Version history

  1. v1Submitted by אלון שכטרInitial depositCurrentOct 25, 2026