bibkey: averkovvondichtersoprunov2026logsubmodularity authors: Gennadiy Averkov; Katherina von Dichter; Ivan Soprunov year: 2026 title: “On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube” doi: null url: https://arxiv.org/html/2608.14909v1#S4.E16 claim: “Equation (16) is the Hypercube Inequality for nonnegative weights on the vertices of the d-dimensional cube; Section 7.5 states that it remains open for d >= 4.” strata_touched:

• D5/S0/FiniteGeometry/HypercubeInequality license: CC BY 4.0 triage: anchor

The Hypercube Inequality

Averkov, von Dichter and Soprunov reduce their proposed log-submodularity inequality for zonoids to the Hypercube Inequality. Section 4.4, Equation (16), states that for nonnegative weights indexed by the vertices of the d-dimensional cube,

(sum over (d+1)-vertex subsets S of Vol(S) times the product of the weights on S) times (the sum of all weights)^(d-1) is at most the product, over all cube facets F, of the sum of the weights on F.

Here Vol(S) is normalized simplex volume. With the columns (v,1), this is the absolute value of the corresponding integer determinant. Each unordered subset occurs once and there is no factorial factor. Pairing the two facets perpendicular to each coordinate gives exactly the two facet sums in the formal result.

The paper proves the inequality for d=3, with lower dimensions included in its discussion. Section 7.5 explicitly says that for d>=4 the Hypercube Inequality remains open. The repository theorem addresses the determinant-defined Equation (16) uniformly for every d>=1; it does not formalize the paper’s geometric reduction or assert a theorem about zonoid volume.

Verified locator

• Primary version: https://arxiv.org/html/2608.14909v1
• Exact cited URL: https://arxiv.org/html/2608.14909v1#S4.E16
• arXiv identifier and submission date: 2608.14909v1, 14 August 2026.
• Manuscript date printed in the source: August 24, 2026.
• Exact target: Section 4.4, Equation (16), HTML anchor S4.E16.
• Open-status statement: Section 7.5, “Limitations for higher dimensions”.
• Source file read locally with SHA-256 b439f4eab795c6e2091720f5a2025616231035b9eaffa0c9d5158b8228fd8d4f.

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