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    "title": "Fenchel establishes symmetric convex conjugacy in finite dimensions",
    "summary": "For a closed proper convex function on finite-dimensional Euclidean space, Fenchel's supremum construction gives a convex conjugate satisfying <x,xi> <= f(x)+f*(xi), with symmetric recovery of the original function by conjugation. Supporting hyperplanes connect equality to conjugate points without requiring differentiability.",
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        "title": "On Conjugate Convex Functions (Canadian Journal of Mathematics 1, 73–77)",
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            "citation": "On Conjugate Convex Functions (Canadian Journal of Mathematics 1, 73–77)",
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        "description": "The original formulates a finite convex function on a convex domain, lower semicontinuity and blow-up at excluded boundary points; the extended-real closed-proper formulation here encodes those domain conditions. The conjugate is finite only on its effective domain, not necessarily on all Euclidean space. Equality need not be attained at every boundary point. The general conjugacy theorem is not an unrestricted smooth Legendre-gradient bijection, an infinite-dimensional theorem, or a strong-duality theorem without qualification.",
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          "citation": "On Conjugate Convex Functions (Canadian Journal of Mathematics 1, 73–77)",
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      "statement": "# Fenchel establishes symmetric convex conjugacy in finite dimensions\n\nFor a closed proper convex function on finite-dimensional Euclidean space, Fenchel's supremum construction gives a convex conjugate satisfying <x,xi> <= f(x)+f*(xi), with symmetric recovery of the original function by conjugation. Supporting hyperplanes connect equality to conjugate points without requiring differentiability.\n\n## Scientific scope\n\nThe original formulates a finite convex function on a convex domain, lower semicontinuity and blow-up at excluded boundary points; the extended-real closed-proper formulation here encodes those domain conditions. The conjugate is finite only on its effective domain, not necessarily on all Euclidean space. Equality need not be attained at every boundary point. The general conjugacy theorem is not an unrestricted smooth Legendre-gradient bijection, an infinite-dimensional theorem, or a strong-duality theorem without qualification.\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\nWerner Fenchel\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- [On Conjugate Convex Functions (Canadian Journal of Mathematics 1, 73–77)](https://www.cs.cmu.edu/~suvrit/teach/papers/1949_fenchel_conjugate_convex_functions.pdf)\n",
      "summary": "For a closed proper convex function on finite-dimensional Euclidean space, Fenchel's supremum construction gives a convex conjugate satisfying <x,xi> <= f(x)+f*(xi), with symmetric recovery of the original function by conjugation. Supporting hyperplanes connect equality to conjugate points without requiring differentiability.",
      "title": "Fenchel establishes symmetric convex conjugacy in finite dimensions"
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    "id": "fenchel-finite-dimensional-conjugate-convex-functions-1949",
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    "title": "Fenchel establishes symmetric convex conjugacy in finite dimensions",
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    "what_changed": "For a closed proper convex function on finite-dimensional Euclidean space, Fenchel's supremum construction gives a convex conjugate satisfying <x,xi> <= f(x)+f*(xi), with symmetric recovery of the original function by conjugation. Supporting hyperplanes connect equality to conjugate points without requiring differentiability.",
    "scope": "For a closed proper convex function on finite-dimensional Euclidean space, Fenchel's supremum construction gives a convex conjugate satisfying <x,xi> <= f(x)+f*(xi), with symmetric recovery of the original function by conjugation. Supporting hyperplanes connect equality to conjugate points without requiring differentiability. The original formulates a finite convex function on a convex domain, lower semicontinuity and blow-up at excluded boundary points; the extended-real closed-proper formulation here encodes those domain conditions. The conjugate is finite only on its effective domain, not necessarily on all Euclidean space. Equality need not be attained at every boundary point. The general conjugacy theorem is not an unrestricted smooth Legendre-gradient bijection, an infinite-dimensional theorem, or a strong-duality theorem without qualification.",
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