A subset C ⊆ ℝⁿ is convex iff for every x, y ∈ C and λ ∈ [0, 1] the point λx + (1−λ)y lies in C. Equivalently, C contains every line segment joining any two of its points. Convex hulls, separating-hyperplane theorems, and the…
Convex set
Related concepts
- Vector space
- Convex function: epigraph convex ⇔ midpoint inequality ⇔ Hessian ≽ 0
- Convex polytope: vertex / half-space duality + Euler V−E+F=2
- Supporting hyperplane + Hahn-Banach separation of convex sets
- Brunn-Minkowski R², A=B=[0,1]²: LHS=RHS=2, equality case
- John ellipsoid (1948)
- Blaschke-Santalo inequality
- Dvoretzky theorem (1961)
- Minkowski (1896)
- Brunn-Minkowski (1887)
- Blaschke selection (1916)
- John ellipsoid (1948)
- Santaló inequality (1949)
- Choquet theorem (Choquet 1956)
- Holevo bound chi = S(rho-bar) - sum p_i S(rho_i) >= I_acc; concavity on convex hull