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    "title": "Loomis–Whitney: volume bounded by coordinate projections",
    "summary": "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.",
    "boundary": "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.",
    "area": "Geometric and combinatorial inequalities",
    "topic": "Mathematics",
    "year": 1949,
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        "title": "An inequality related to the isoperimetric inequality (original1949)",
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        "description": "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.",
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          "citation": "An inequality related to the isoperimetric inequality (original1949)",
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      "statement": "# Loomis–Whitney: volume bounded by coordinate projections\n\nLoomis 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.\n\n## Scientific scope\n\nThe 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nL. H. Loomis; H. Whitney\n\nSource-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.\n\n\n\n## Original sources\n\n- [An inequality related to the isoperimetric inequality (original1949)](https://www.ams.org/journals/bull/1949-55-10/S0002-9904-1949-09320-5/S0002-9904-1949-09320-5.pdf)\n",
      "summary": "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.",
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    "id": "loomis-whitney-volume-projection-inequality-1949",
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    "title": "Loomis–Whitney: volume bounded by coordinate projections",
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    "what_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.",
    "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_summary": "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.",
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