striktConvexe

striktConvexe, often written strictly convex in English, is a property in convex analysis that can apply to functions or to sets, indicating a stronger form of convexity.

When applied to a function, striktConvexe means that the value of the function lies strictly below the chord between any two distinct points in its domain. Formally, a function f defined on a convex domain D is strictly convex if for all x and y in D with x ≠ y and all t in (0,1), f(t x + (1−t) y) < t f(x) + (1−t) f(y). If f is differentiable, this is equivalent to the condition that the line segment between any two points lies above the graph of f, and, in the twice differentiable case, to the Hessian being positive definite on D. Strict convexity implies the minimizer of f on a convex set, when it exists, is unique.

When striktConvexe is a property of a set, a subset C of a vector space is strictly convex if for any distinct x,y in C and any t in (0,1), the point t x + (1−t) y lies in the interior of C. Intuitively, the line segment between any two points of C is contained entirely inside C, not merely on its boundary. Examples include Euclidean balls; polygons with curved boundaries or no flat edges are strictly convex, while polygons with flat sides are not.

Key consequences include the uniqueness of minimizers for strict convex optimization problems and, for smooth f, a strict positive curvature condition (positive definite Hessian) that strengthens standard convexity. StriktConvexe plays a central role in optimization theory, economics, and numerical algorithms, where uniqueness and stability of solutions are desirable.