Validity of the projection-volume ratio inequality (7) for zonoids in higher dimensions

Determine whether the inequality |A| / |P_E A| ≤ |A + B| / |P_E(A + B)| holds for all pairs of zonoids A,B ⊂ R^n and all hyperplanes E when n ≥ 4. Establish either a proof of the inequality for zonoids in dimensions n ≥ 4 or provide counterexamples, thereby resolving its validity in higher dimensions.

Background

The authors discuss a conjectured concavity property comparing the ratio of volumes to volumes of projections, and a weaker form given by inequality (7). While inequality (7) fails in general for convex bodies in dimensions n ≥ 3, it has been shown to hold for the class of zonoids when n = 3.

The status of inequality (7) for zonoids in higher dimensions is currently unknown, and resolving this would clarify how far the special structure of zonoids supports the projection-volume ratio bound beyond the three-dimensional case.

References

It has been demonstrated in [20] that (7) is valid for the class of zonoids when n = 3 (the case of the zonoids in higher dimensions is still open).

— On the volume of sums of anti-blocking bodies  (2409.14214 - Manui et al., 2024) in Introduction, after equation (7)

The higher-dimensional weak problem remains open in the intermediate range. We conjecture that it is affirmative for 1<p<2 in dimension three and negative for every 1<p<2 in dimensions n\geq4.

— Volume and Projection Inequalities II: Determinants and $L_p$-Sums  (2608.24081 - Fradelizi et al., 25 Aug 2026) in Section 1, subsection “Inequalities for volume” (immediately after Theorem 1.3 / Theorem \ref{thm:ratio-Lp-zonoidwekk})