AWhat changed
Loomis and Whitney prove that the volume m of an open subset of R^n satisfies m^(n−1)≤m_1⋯m_n, where m_i is its (n−1)-dimensional coordinate projection measure. Their elementary induction and Hölder argument first establishes the corresponding cube-counting inequality N^(n−1)≤N_1⋯N_n. Axis-parallel boxes attain equality in the volume bound.
Scientific scope
Loomis and Whitney prove that the volume m of an open subset of R^n satisfies m^(n−1)≤m_1⋯m_n, where m_i is its (n−1)-dimensional coordinate projection measure. Their elementary induction and Hölder argument first establishes the corresponding cube-counting inequality N^(n−1)≤N_1⋯N_n. Axis-parallel boxes attain equality in the volume bound. The original theorem concerns open Euclidean sets and coordinate-hyperplane projections; its discrete step counts cubes in a common cubical subdivision. The derived surface-area inequality lacks the optimal isoperimetric constant. Do not replace this by a reverse projection inequality, claim equality only for a class not classified in the paper, or assign all later functional, entropy, nonorthogonal or nonlinear Brascamp–Lieb generalizations to this note.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original theorem concerns open Euclidean sets and coordinate-hyperplane projections; its discrete step counts cubes in a common cubical subdivision. The derived surface-area inequality lacks the optimal isoperimetric constant. Do not replace this by a reverse projection inequality, claim equality only for a class not classified in the paper, or assign all later functional, entropy, nonorthogonal or nonlinear Brascamp–Lieb generalizations to this note.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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