Convex and strictly convex functions on Euclidean convex sets

Depends on

• A convex subset of $\mathbb{R}^m$ contains every line segment between two of its points
• Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length

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
• Squared distance is strictly convex along geodesics in a hadamard manifold Corollary
• The Lax shock inequalities for convex scalar laws 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
• Convex entropy--entropy flux pairs Definition
• Subgradients and the subdifferential of a convex function Definition
• The Legendre transform of a finite-valued convex Hamiltonian 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 finite-valued convex Hamiltonian equals its biconjugate Lemma
• The convex entropy condition for a single shock is the chord condition 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
• The Hopf--Lax operators form a semigroup (dynamic programming) Theorem
• The Riemann solver for a strictly convex flux 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)