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Isoperimetric inequalities derived from affine quermassintegrals in complex and quaternionic spaces

Prove that for every convex body K in F^n with non-empty interior, where F ∈ {C, H} and for every integer m with 1 ≤ m ≤ n, the isoperimetric inequality (κ_{np} κ_{mp}) ∫_{Gr_m(n,F)} |P_E K| dE ≥ |K|^{n/m} holds.

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Background

The conjectured affine quermassintegral inequality implies a family of isoperimetric inequalities relating the average projected volume of a convex body to its volume. In the real case, these inequalities follow from the confirmed conjecture; however, in the complex and quaternionic settings they are not yet established.

The authors emphasize that standard symmetrization tools used in the real case are not directly applicable over C and H, which contributes to the unresolved status of these isoperimetric inequalities.

References

As in the real case, the conjecture directly implies the isoperimetric inequalities \begin{equation}\label{eq:iso}\frac{\kappa_{np}{\kappa_{mp} \int_{Gr_m(n,F)} |P_E K| dE \geq |K|{n/m}.\end{equation} While these inequalities are highly compelling, they remain open over the complex numbers and the quaternions.

eq:iso:

κnpκmp∫Grm(n,F)∣PEK∣dE≥∣K∣n/m.\frac{\kappa_{np}}{\kappa_{mp}} \int_{Gr_m(n,F)} |P_E K| dE \geq |K|^{n/m}.

— Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalities (2409.01057 - Saroglou et al., 2 Sep 2024) in Equation (eq:iso) and following sentence, Introduction