Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Transfer of Approximating Test Data from the Sup-Convolution to the Original Function
lemmalem:sup-convolution-test-data-transfer-2026aAnalysisPDEShows that test data approximable from above for the sup-convolution at a point with first-order datum transfers to the same first-order and second-order data for itself at , together with the accompanying identity for the…Concatenation and the Sup-Convolution of a Sum in Separated Variables
lemmalem:sup-convolution-separate-variables-2026aAnalysisMultivariable CalculusRecords that concatenation carries origins to the origin and is compatible with coordinatewise convergence, and shows that the sup-convolution of a function of the form is the sum of the sup-convolutions of and with the same parameter.Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation
settingset:euclidean-calculus-2026aAnalysisMultivariable CalculusFixes the standing notation for partial derivatives, functions of class , gradients and Hessians, twice differentiability at a point in the second-order expansion sense, continuity, semicontinuity and local extrema, and convex and semiconvex functions on Euclidean space.Localisation of an Upper Semicontinuous Function by Projection onto a Closed Ball
lemmalem:usc-ball-localisation-2026aAnalysisMultivariable CalculusDescribes the nearest point projection onto a closed ball centred at the origin and shows that composing an upper semicontinuous function with it, minus a multiple of the excess of the squared norm over the squared radius, produces an upper semicontinuous function on all of Eucli…Quadratic Test Functions, Limits, Translation and Locality for Approximability by Test Data
lemmalem:test-data-basic-2026aAnalysisPDEApproximability by test data can always be realised with quadratic test functions; it follows at a point of twice differentiability, is stable under limits of the approximating data, transforms predictably under translation and affine perturbation, and depends only on the values…Twice Differentiability of a Sum in Separated Variables
lemmalem:twice-differentiable-separated-2026aAnalysisMultivariable CalculusA sum in separated variables is twice differentiable at a point exactly when both summands are, and its Hessian is then the block diagonal matrix built from their Hessians; in particular the off-diagonal block vanishes.Quadratic and Affine Functions of Class , Translation, and Quadratic Perturbation of Semiconvexity
lemmalem:quadratic-affine-c2-2026aAnalysisMultivariable CalculusThe function obtained from a symmetric matrix, a vector and a constant is of class with the expected gradient and Hessian; class and its derivatives are preserved by translation; positive semidefinite quadratic forms are convex; and subtracting a quadratic form from a…Limits and Bounded Sequences of Symmetric Real Matrices
lemmalem:symmetric-matrix-limits-2026aAnalysisLinear AlgebraA norm-bounded sequence of symmetric real matrices has a convergent subsequence; quadratic forms depend continuously on the matrix; the positive semidefinite ordering passes to limits; and two-sided order bounds give a norm bound.Sums, Differences and Scalar Multiples of Functions Twice Differentiable at a Point
lemmalem:twice-differentiable-sum-2026aAnalysisMultivariable CalculusTwice differentiability at a point, in the second-order expansion sense, is preserved by sums, differences and scalar multiples, with the first-order coefficients and Hessians combining in the same way.A Weighted Young Inequality and the Splitting of a Quadratic Form
lemmalem:quadratic-form-splitting-2026aAnalysisLinear AlgebraRecords the weighted Young inequality for the dot product and deduces the inequality comparing the quadratic form of a symmetric matrix at a point with its value at a second point, with a squared-distance penalty.Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set
corollarycor:alexandrov-open-semiconvex-rn-2026aAnalysisMultivariable CalculusA semiconvex function with constant on an open convex subset of Euclidean space is twice differentiable outside a null set, and wherever it is twice differentiable its Hessian is bounded below by .Alexandrov's Theorem: a Convex Function on is Twice Differentiable Almost Everywhere
theoremthm:alexandrov-convex-rn-2026aAnalysisMultivariable CalculusA convex function on Euclidean space admits a second-order expansion with a symmetric Hessian at every point outside a Borel null set, and wherever it is twice differentiable its Hessian is positive semidefinite.A First-Order Expansion of the Subdifferential Gives a Second-Order Expansion of the Function
lemmalem:subgradient-expansion-twice-differentiable-rn-2026aAnalysisMultivariable CalculusIf the subgradients of a convex function admit a first-order expansion about a point with matrix , then the function admits a second-order expansion there with Hessian the symmetric part of ; this is obtained by telescoping the subgradient inequalities along a segment.Basic Properties of Twice Differentiability at a Point
lemmalem:twice-differentiable-basic-rn-2026aAnalysisMultivariable CalculusAt a point of twice differentiability the first-order coefficient is the gradient; a function of class is twice differentiable at every point, with the usual gradient and Hessian; and at a local maximum the gradient vanishes and the Hessian is negative semidefinite.Twice Differentiability at a Point
definitiondef:twice-differentiable-at-point-rn-2026aAnalysisMultivariable CalculusDefines twice differentiability of a real-valued function at a point of an open subset of Euclidean space, by a second-order expansion with a symmetric Hessian, the coefficients being unique.An Injective Nondegenerate Square Matrix is Invertible
lemmalem:square-matrix-injective-invertible-rn-2026aAnalysisLinear AlgebraIf the map determined by a real square matrix is injective it satisfies a lower bound and has closed convex image; if in addition no unit vector annihilates that image, the matrix is surjective and hence invertible, with an inverse obeying the reciprocal bound.Extending a Convex Function from a Closed Ball to All of
lemmalem:convex-extension-from-ball-rn-2026aAnalysisMultivariable CalculusThe supremum of the affine minorants furnished by the subgradients at points of a closed ball is a convex Lipschitz function on all of Euclidean space that agrees with the given convex function on that ball.Properties of the Proximal Map of a Convex Function
lemmalem:proximal-map-properties-rn-2026aAnalysisMultivariable CalculusThe proximal map of a convex function has fibres described by the subdifferential, is surjective, is firmly nonexpansive and hence Lipschitz with constant one, and wherever it is differentiable its derivative matrix satisfies and is injectiv…The Proximal Map of a Convex Function on
definitiondef:proximal-map-convex-rn-2026aAnalysisMultivariable CalculusDefines the proximal map of a convex function on Euclidean space, sending a point to the unique minimiser of .Existence and Uniqueness of the Proximal Minimiser of a Convex Function
lemmalem:proximal-minimiser-rn-2026aAnalysisMultivariable CalculusFor a convex function on Euclidean space and any point , the function attains its minimum at exactly one point.