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Makai Jr.'s reverse isodiametric conjecture

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Let Kn\mathcal{K}^n be the family of full-dimensional convex compact sets in Rn\mathbb{R}^n, and let Kon\mathcal{K}^n_o denote the oo-symmetric members. For K∈KnK\in\mathcal{K}^n, define its diameter by D(K)=max⁡{∥x−y∥:x,y∈K}\mathrm{D}(K)=\max\{\|x-y\|:x,y\in K\} and its isodiametric quotient by

idq⁡(K)=vol(K)D(K)n.\operatorname{idq}(K)=\frac{\mathrm{vol}(K)}{\mathrm{D}(K)^n}.

A regular simplex and a regular crosspolytope are the corresponding standard comparison bodies.

Makai Jr.'s conjecture. For every K∈KnK\in\mathcal{K}^n there is a linear transformation A∈GL⁡n(R)A\in\operatorname{GL}_n(\mathbb{R}) such that

idq⁡(AK)≥n+1n! 2n2,\operatorname{idq}(AK)\geq\frac{\sqrt{n+1}}{n!\,2^{\frac n2}},

with equality if and only if AKAK is a regular simplex. If K∈KonK\in\mathcal{K}^n_o, then there is an A∈GL⁡n(R)A\in\operatorname{GL}_n(\mathbb{R}) such that

idq⁡(AK)≥1n!,\operatorname{idq}(AK)\geq\frac1{n!},

with equality if and only if AKAK is a regular crosspolytope.

The conjecture concerns the reverse isodiametric problem and is motivated by applications to the minimal density of non-separable lattice arrangements of convex bodies. It is open for dimensions n≥3n\geq3, while the reverse isodiametric problem is solved in the plane.

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  1. Makai Jr.'s reverse isodiametric conjecture

    Let K⊂RnK\subset\mathbb{R}^n be a convex body, let idq(K)=Vol⁡(K)Diam⁡(K)nidq(K)=\frac{\operatorname{Vol}(K)}{\operatorname{Diam}(K)^n} be its isodiametric quotient, and let TT be a linear bijection of Rn\mathbb{R}^n. A regular simplex in Rn\mathbb{R}^n has isodiametric quotient n+1n!2n/2\frac{\sqrt{n+1}}{n!2^{n/2}}. Makai Jr.'s conjecture. For every convex body K⊂RnK\subset\mathbb{R}^n, there exists a linear bijection TT such that

    idq(TK)≥n+1n!2n/2.idq(TK)\geq\frac{\sqrt{n+1}}{n!2^{n/2}}.

    This is a reverse form of the isodiametric problem: after optimizing the diameter by a volume-preserving linear transformation, every convex body should attain at least the quotient of a regular simplex. The supplied source gives no evidence of resolution.

    source: Arkadiy Aliev, “The exact bound for the reverse isodiametric problem in 3-space”, arXiv:2306.14576 (2023).

References

Primary source

Bernardo González Merino and Matthias Schymura, “On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds”, arXiv:1804.05009 (2020).

convex-geometryisodiametric-inequalities
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