What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
L. H. Loomis; H. Whitney
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English catalogue narrative
Loomis–Whitney: volume bounded by coordinate projections
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
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.
Reading status
This 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.
People and systems
L. H. Loomis; H. Whitney
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.