SWhat 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.
Scientific 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.
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 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.
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.
Full scientific record · Browse contribution grades · Request a correction