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…
Helly's theorem
Helly 1923: if every d+1 of a finite collection of convex sets in ℝ^d have common point, all do. Color/fractional/topological Helly…
Carathéodory's theorem
Every point in convex-hull of S ⊂ ℝ^d lies in convex-hull of at most d+1 points of S. Refinement by Steinitz. Carathéodory number d+1 of…
Radon's partition theorem
Radon 1921: any d+2 points in ℝ^d can be partitioned into two sets whose convex hulls intersect. Bridges Helly and Carathéodory. Tverberg…
Minkowski sum & difference
A ⊕ B = {a+b : a∈A, b∈B}; A ⊖ B = {x : x+B ⊆ A}. Brunn-Minkowski inequality: vol(A⊕B)^(1/d) ≥ vol(A)^(1/d) + vol(B)^(1/d). Foundational…
Krein-Milman (extreme points)
Krein-Milman 1940: compact convex set in locally-convex topological vector space equals closed-convex-hull of its extreme points. Choquet…
Blaschke selection theorem
Blaschke 1916: every uniformly-bounded sequence of convex bodies in ℝ^d has a Hausdorff-convergent subsequence. Compactness of…
Brunn-Minkowski inequality (1887/1896)
Brunn 1887 / Minkowski 1896: vol(A+B)^(1/n) >= vol(A)^(1/n) + vol(B)^(1/n) for convex A,B in R^n; foundational for isoperimetry + entropy.
John ellipsoid (1948)
F John 1948: every convex body K contains a unique ellipsoid of maximal volume; if K is symmetric, John ellipsoid scaling factor sqrt(n)…
Blaschke-Santalo inequality
Blaschke 1917 + Santalo 1949: for symmetric convex K, vol(K) vol(K^*) <= vol(B)^2 with equality iff K is ellipsoid; Bourgain-Milman lower…
Dvoretzky theorem (1961)
Dvoretzky 1961: every n-dim normed space has subspace of dimension d ~ log n that is (1+eps)-isomorphic to Euclidean ball; Milman 1971…
Polytope face structure (Euler 1758)
Euler 1758 V - E + F = 2 for 3-polytopes; Schlafli 1850 generalization to any d; basis of polyhedral combinatorics + simplicial complex…
Kahn-Saks comparison (1/3-2/3 conjecture)
Kahn-Saks 1984 / Brightwell-Felsner-Trotter 1995: in any finite poset, exists pair (x,y) with P(x<y) in [1/3, 2/3]; conjectured 1/3-2/3…
Minkowski (1896)
H Minkowski 1896 'Geometrie der Zahlen'; modern foundational text on convex-bodies + lattice-points + geometry-of-numbers.
Brunn-Minkowski (1887)
H Brunn 1887 + Minkowski 1903 V(K+L)^{1/n} ≥ V(K)^{1/n}+V(L)^{1/n}; modern functional-BL Prékopa-Leindler 1972.
Blaschke selection (1916)
W Blaschke 1916 selection-theorem compact-convex-sets; modern Hausdorff metric + isoperimetric + Loomis-Whitney inequality.
John ellipsoid (1948)
F John 1948 maximal-volume ellipsoid; modern Banach-Mazur distance + Bourgain-Milman + restricted-isometry + geometric-functional-analysis.
Santaló inequality (1949)
L Santaló 1949 V(K)V(K°) ≤ V(B_n)²/n!; modern Mahler-conjecture-still-open + Bourgain-Milman lower-bound 1987.
Choquet theorem (Choquet 1956)
G Choquet 1956 + Bishop-de-Leeuw 1959 extreme-point integral-representation; modern measure-theoretic convex analysis.
Bourgain-Milman reverse Santalo inequality |K| |K^*| >= c^n / n!
Bourgain-Milman inequality (Bourgain-Milman 1987 *Inventiones* 88, 319) is the asymptotic reverse of the Blaschke-Santalo inequality: for…