HEXAGONbeta

Illumination of several families of complex convex bodies

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

Contributed by אלון שכטר ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We determine ordinary and fractional illumination numbers for complex Hanner bodies and for bodies obtained as convex hulls of coordinate polydiscs indexed by the maximal independent sets of a graph. The formulas are sums over coordinate supports. For Hanner bodies we classify all equality cases in the ordinary and fractional dimension bounds, including the sharp ordinary bound for non-polydiscs. For graph bodies, fractional equality 2n2^n occurs exactly when the graph is a disjoint union of isolated vertices and edges; a vertex of degree at least two gives the strict bound 3⋅2n−23\cdot2^{n-2}. We also prove the fractional bound 2n2^n for bodies defined by nondecreasing submodular constraints on coordinate moduli. For finite systems of positive weighted inequalities satisfying an explicit condition on tight sets, we obtain the same fractional bound and an ordinary bound of (2+o(1))nlog⁡n 2n(2+o(1))n\log n\,2^n, together with an explicit estimate in every dimension. This includes bounded systems on laminar families with arbitrary positive coefficients. We conclude with a conjectural classification of fractional equality for normalized complex 11-unconditional bodies.

Provenance statement

Results generated with Chatgpt Astra-6. Hanner bodies results have been checked by a human

References

  1. E. D. Baladze and V. G. Boltyanski, Illumination of direct sums of two convex figures , Beitr. Algebra Geom. 47 (2006), no. 2, 345--350. Full text (https://ftp.gwdg.de/pub/misc/EMIS/journals/BAG/vol.47/no.2/b47h2bal.pdf).link
  2. V. Boltyanski and H. Martini, Illumination of direct vector sums of convex bodies , Studia Sci. Math. Hungar. 44 (2007), no. 3, 367--376. doi:10.1556/SScMath.44.2007.3.6 (https://doi.org/10.1556/SScMath.44.2007.3.6).DOI
  3. V. Chvátal, On certain polytopes associated with graphs , J. Combin. Theory Ser. B 18 (1975), no. 2, 138--154. doi:10.1016/0095-8956(75)90041-6 (https://doi.org/10.1016/0095-8956(75)90041-6).DOI
  4. L. Danzer and B. Grünbaum, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee , Math. Z. 79 (1962), 95--99. doi:10.1007/BF01193107 (https://doi.org/10.1007/BF01193107).DOI
  5. J. Edmonds, Submodular functions, matroids, and certain polyhedra , in Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf., 1969), Gordon and Breach, New York, 1970, 69--87. Reprinted in Combinatorial Optimization---Eureka, You Shrink! , Lecture Notes in Comput. Sci. 2570 , Springer, 2003, 11--26. doi:10.1007/3-540-36478-1_2 (https://doi.org/10.1007/3-540-36478-1_2) (reprint).DOI
  6. O. Hanner, Intersections of translates of convex bodies , Math. Scand. 4 (1956), 65--87. doi:10.7146/math.scand.a-10456 (https://doi.org/10.7146/math.scand.a-10456).DOI
  7. J. Kim, Minimal volume product near Hanner polytopes , J. Funct. Anal. 266 (2014), no. 4, 2360--2402. doi:10.1016/j.jfa.2013.08.008 (https://doi.org/10.1016/j.jfa.2013.08.008). arXiv:1212.2544 (https://arxiv.org/abs/1212.2544).arXivDOI
  8. F. Kohl, M. Olsen, and R. Sanyal, Unconditional reflexive polytopes , Discrete Comput. Geom. 64 (2020), 427--452. doi:10.1007/s00454-020-00199-8 (https://doi.org/10.1007/s00454-020-00199-8).DOI
  9. L. Lovász, On the ratio of optimal integral and fractional covers , Discrete Math. 13 (1975), no. 4, 383--390. doi:10.1016/0012-365X(75)90058-8 (https://doi.org/10.1016/0012-365X(75)90058-8).DOI
  10. M. Naszódi, Fractional illumination of convex bodies , Contrib. Discrete Math. 4 (2009), no. 2, 83--88. doi:10.55016/ojs/cdm.v4i2.62022 (https://doi.org/10.55016/ojs/cdm.v4i2.62022).DOI
Show all 13 referencesShow fewer references
  1. S. Reisner, Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases , J. London Math. Soc. (2) 43 (1991), no. 1, 137--148. doi:10.1112/jlms/s2-43.1.137 (https://doi.org/10.1112/jlms/s2-43.1.137).DOI
  2. 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). arXiv:2410.12021 (https://arxiv.org/abs/2410.12021).arXivDOI
  3. Illumination of complex 11-unconditional convex bodies , unpublished companion working manuscript, version of 25 October 2026, Section 7, ``Fractional illumination in two complex dimensions.''

Version history

  1. v1Submitted by אלון שכטרInitial depositCurrentNov 01, 2026