Geometry of convex sets and convex functions in Euclidean (and more generally Banach) spaces. Foundational objects: convex sets (closed under line segments), convex hulls (smallest convex enclosure of a set), convex functions (epigraph is…
convex-geometry
Convex function: epigraph convex ⇔ midpoint inequality ⇔ Hessian ≽ 0
A function f : C → R on a convex set C is convex iff f(λx + (1−λ)y) ≤ λf(x) + (1−λ)f(y) for all x, y ∈ C and λ ∈ [0, 1]. Four equivalent…
Convex polytope: vertex / half-space duality + Euler V−E+F=2
A convex polytope P ⊂ R^n admits two equivalent descriptions: (V-description) P is the convex hull of finitely many points v_1, …, v_k —…
Supporting hyperplane + Hahn-Banach separation of convex sets
Supporting hyperplane theorem: at every boundary point x₀ of a convex set C in a normed space there exists a supporting hyperplane H = {x :…
Jensen gap for f=x²: (x₁−x₂)²/4 symbolic; (1,3) pin = 1; diag = 0
Exact sympy witness that x ↦ x² is strictly convex: the two-point Jensen gap (f(x₁) + f(x₂))/2 − f((x₁ + x₂)/2) factors as (x₁ − x₂)²/4…
Brunn-Minkowski R², A=B=[0,1]²: LHS=RHS=2, equality case
Sympy witness for the equality case of the Brunn-Minkowski inequality in R². Setup: A = B = [0, 1]² unit cubes, so A + B = [0, 2]²…
Cube Euler χ: V−E+F = 8−12+6 = 2 = χ(S²)
Sympy-exact arithmetic witness of Euler's 1752 polytope formula for the cube. Face counts: V = 8 vertices, E = 12 edges, F = 6 square…