Alphabeta MathHow statement and proof provenance work
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Convex and strictly convex functions on Euclidean convex sets
Definition
Let be convex (A convex subset of contains every line segment between two of its points). The function is convex when for all and . The parameter interval is the closed interval of Intervals of : the nine order-convex forms, nondegeneracy, and length.
It is strictly convex when the inequality is strict for distinct and . No strict inequality is required at or , where the two sides coincide. The empty set and a singleton support convex functions, and strict convexity on either is vacuous.
Depends on
Used by
- A strictly convex function on a Euclidean convex set has at most one global minimizer Corollary
- Every local minimum of a convex function on an open Euclidean convex set is global Corollary
- The subdifferential of a differentiable convex function is its gradient singleton Corollary
- A convex function on [0,1] that is discontinuous at the boundary Counterexample
- A positive-semidefinite Hessian need not give strict convexity Counterexample
- A strictly convex function can have a singular Hessian Counterexample
- x↦-√1-‖x‖₂² has empty subdifferential on the unit-sphere boundary Counterexample
- Subgradients and the subdifferential of a convex function Definition
- The Euclidean norm and its square are convex, with a ball of subgradients at zero for the norm Example
- A convex function is bounded above and below on a smaller interior cube Lemma
- A pointwise supremum of convex functions is convex wherever it is finite Proposition
- Every sublevel set of a convex function is convex Proposition
- Nonnegative combinations, affine precomposition, and finite pointwise maxima preserve convexity Proposition
- A C² function is convex exactly when its Hessian is positive semidefinite Theorem
- A convex function on an open convex set is locally Lipschitz Theorem
- A function is convex exactly when its epigraph is convex Theorem
- Differentiable convex functions are characterized by the gradient inequality Theorem
- Finite Jensen inequality for convex functions on ℝⁿ Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Boyd and L. Vandenberghe, Convex Optimization, §3.1.1 (standard reference, not scraped)
- D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 2 (standard reference, not scraped)