Convex sets and functions

A set is convex when the entire line segment joining any two of its points stays inside the set. A disk, a half-space, and an affine plane pass this test. The exterior of a disk fails: two allowed points on opposite sides have a forbidden midpoint. Intersections preserve convexity because a segment lying in every constituent set also lies in their intersection. Unions generally do not have this property.

A convex function is defined on a convex domain and lies below each chord joining two points of its graph. This definition compares function values, not slopes, so it includes nonsmooth functions such as absolute value. A convex optimization problem minimizes a convex objective over a convex feasible set. In the standard inequality description, convex constraint functions and affine equalities provide a useful sufficient construction; arbitrary nonlinear equalities need separate geometric checking.

For a differentiable convex function on an open convex domain, its tangent plane is a global lower bound. Thus a zero gradient certifies a global minimizer without requiring a search through every other point. Under constraints, the gradient need not vanish: it can point toward forbidden locations. The corresponding first-order certificate asks whether its inner product with every feasible displacement is nonnegative. Convexity then turns that directional statement into a value comparison.

For twice continuously differentiable functions on an open convex domain, positive semidefiniteness of the Hessian everywhere characterizes convexity. Testing the Hessian at one candidate only describes local curvature. A function may have a positive Hessian near one minimum and a deeper minimum elsewhere. Likewise, a collection of sampled nonnegative second derivatives is evidence about the samples, not a proof about all unsampled points of a continuous domain.

Strict convexity replaces the chord inequality by a strict inequality for distinct points and interior interpolation weights. It ensures that a convex feasible set has at most one minimizer, but does not make a minimizer exist. The strictly convex function $e^x$ on the real line has infimum zero and never attains it. Existence follows, for example, from continuity on a nonempty compact feasible set. In finite dimensions, a lower-semicontinuous coercive objective on a nonempty closed feasible set also attains its infimum, provided at least one feasible point has finite value.

The interactive quadratic is one particularly friendly convex problem, not a test of arbitrary objective functions. Change its initial position and ask whether the destination depends on which side you start from. Then change the step size beyond two: divergence reflects an unstable algorithm, despite the objective remaining convex with a unique minimizer. Geometry of the problem and behavior of a chosen iteration are different claims that should be reported separately.