Lecture 3: Convex Hull Characterization and Gauss-Lucas Theorem

This file formalizes Proposition 1 from Lecture 3, which characterizes the convex hull of a finite set of points in ℂ as the set of all convex combinations, and the Gauss-Lucas theorem (Theorem 1): the smallest convex set containing all the zeros of a polynomial P(z) with complex coefficients also contains the zeros of P'(z).

Main results

• convexHull_finset_eq_convex_combinations: The convex hull of a finite set S ⊆ ℂ equals {∑ mᵢ aᵢ | mᵢ ≥ 0, ∑ mᵢ = 1}.
• Polynomial.gauss_lucas: The root set of P.derivative is contained in the convex hull (over ℝ) of the root set of P, for any polynomial P of degree ≥ 1.
• Polynomial.gauss_lucas_isRoot: Variant using IsRoot instead of rootSet membership.
• Polynomial.gauss_lucas_centerMass: The convex combination representation: any root z of P' equals a center of mass of the roots of P with explicit nonneg weights that sum to a positive number.

Proof outline (Gauss-Lucas)

The proof follows the textbook argument (Lecture 3, Theorem 1). Write P(z) = aₙ ∏ (z - αᵢ) and take the logarithmic derivative: P'(z)/P(z) = ∑ 1/(z - αᵢ). If z₀ is a zero of P' with P(z₀) ≠ 0, then ∑ 1/(z₀ - αᵢ) = 0. Multiplying the i-th term by conj(z₀ - αᵢ)/conj(z₀ - αᵢ) and taking conjugates shows z₀ = ∑ mᵢ αᵢ where mᵢ = |z₀ - αᵢ|⁻² / ∑ⱼ |z₀ - αⱼ|⁻² are nonneg and sum to 1. Hence z₀ is a convex combination of the αᵢ, and therefore lies in their convex hull. If instead P(z₀) = 0, then z₀ is itself a root of P and hence trivially in the convex hull.

References

• Lecture 3, Proposition 1 and Theorem 1 (Stronger version) of the course on Complex Variables.

theorem convexHull_finset_eq_convex_combinations (S : Finset ℂ) : (convexHull ℝ) ↑S = {x : ℂ | ∃ (w : ℂ → ℝ), (∀ a ∈ S, 0 ≤ w a) ∧ ∑ a ∈ S, w a = 1 ∧ ∑ a ∈ S, w a • a = x}

Proposition 1, Lecture 3. Given a finite set S of points a₁, …, aₙ ∈ ℂ, the set {∑ mᵢ aᵢ | mᵢ ≥ 0, ∑ mᵢ = 1} is the intersection of all convex sets containing every aᵢ, i.e., the convex hull of S.

In other words, convexHull ℝ S equals the set of all convex combinations of elements of S.

theorem Polynomial.gauss_lucas {P : Polynomial ℂ} (hP : 0 < P.degree) : (derivative P).rootSet ℂ ⊆ (convexHull ℝ) (P.rootSet ℂ)

Gauss-Lucas Theorem (Theorem 1, Lecture 3): The smallest convex set containing all the zeros of a polynomial P(z) also contains the zeros of P'(z).

If P is a polynomial over ℂ of degree ≥ 1, then every root of the derivative P' lies in the convex hull (over ℝ) of the roots of P.

theorem Polynomial.gauss_lucas_isRoot {P : Polynomial ℂ} (hP : 0 < P.degree) {z : ℂ} (hz : (derivative P).IsRoot z) : z ∈ (convexHull ℝ) (P.rootSet ℂ)

Variant of the Gauss-Lucas theorem stated with IsRoot: if z is a root of P' and P has positive degree, then z lies in the convex hull of the roots of P.

theorem Polynomial.gauss_lucas_centerMass {P : Polynomial ℂ} (hP : 0 < P.degree) {z : ℂ} (hz : eval z (derivative P) = 0) : z = P.roots.toFinset.centerMass (P.derivRootWeight z) id

The convex combination representation underlying the Gauss-Lucas theorem: if z is a root of P' and P has positive degree, then z equals the center of mass of the roots of P with explicitly defined nonneg weights P.derivRootWeight z.