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Improved and sharp bounds for illumination

Primarily AI-generated textHuman understanding: no partsmath.MG — Metric Geometry

Contributed by אלון שכטר ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove improved ordinary illumination bounds and exact ordinary and fractional values for several classes of convex bodies. Every full-dimensional complex zonotope in $\C^2$ has illumination number five, except complex affine images of the bidisc, whose number is seven. For complex 11-symmetric non-polydiscs in $\C^3$, an explicit construction gives twelve illuminating directions. For real 11-symmetric non-cubes in Rn\R^n, connected domination in the cube graph gives the upper bound $2^{n-1}+\gamma_c(Q_{n-1})+n-1 =2^{n-1}(1+(1+o(1))/n)$; an explicit family has illumination number 2n−12^{n-1}. We also give a product boundary measure proving fractional lower bounds 2n2^n for bodies between specified inner and outer bodies, and attain these bounds along Minkowski interpolations with a positive inner summand. Finally, the least size UnU_n of a single family illuminating every normalized complex 11-unconditional body satisfies Un∼3n+1/πnU_n\sim3^{n+1}/\sqrt{\pi n}.

Provenance statement

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References

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  1. Illumination of several families of complex convex bodies , unpublished companion working research draft, revised 1 November 2026; section on complex Hanner bodies and the final equality conjecture.

Version history

  1. v1Submitted by אלון שכטרInitial depositCurrentNov 01, 2026