<p>We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> is equal to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2^{n+1}-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and that the fractional illumination number of the polydisc in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> is equal to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.</p>

The complex Illumination problem

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2026年1月7日

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Liran Rotem,
Alon Schejter,
Boaz A. Slomka

摘要

We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in $\mathbb {C}^n$ $C^{n}$ , as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in $\mathbb {C}^n$ $C^{n}$ is equal to $2^{n+1}-1$ $2^{n + 1} - 1$ and that the fractional illumination number of the polydisc in $\mathbb {C}^n$ $C^{n}$ is equal to $2^n$ $2^{n}$ . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.