Fixed points on hyperspaces of convex sets

DefinitionHausdorff continuity on compact convex sets#✓fully proved

Let \(E\) be a real normed space, let \(K \subseteq E\), let \(f\) be a map from subsets of \(E\) to subsets of \(E\), and write \(d_H\) for the Hausdorff distance between subsets of \(E\). Then \(f\) is Hausdorff continuous on the compact convex subsets of \(K\) if for every nonempty compact convex set \(X \subseteq K\) and every \(\varepsilon > 0\) there is \(\delta > 0\) such that every nonempty compact convex set \(Y \subseteq K\) with \(d_H(X, Y) < \delta\) satisfies \(d_H(f(X), f(Y)) < \varepsilon\).

Here \(d_H\) is taken to be \(0\) where the Hausdorff distance is infinite, in particular when exactly one of the two sets is empty. The values \(f(X)\) are not required to be compact, convex or nonempty.

def HausdorffContinuousOnConvexCompacts {E : Type} [NormedAddCommGroup E]
    [NormedSpace ℝ E] (f : Set E → Set E) (K : Set E) : Prop :=
  ∀ X : Set E, X ⊆ K → IsCompact X → Convex ℝ X → X.Nonempty → ∀ ε : ℝ, 0 < ε →
    ∃ δ : ℝ, 0 < δ ∧ ∀ Y : Set E, Y ⊆ K → IsCompact Y → Convex ℝ Y → Y.Nonempty →
      Metric.hausdorffDist X Y < δ → Metric.hausdorffDist (f X) (f Y) < ε

Used by 1

Used byFixed point theorem for maps of compact convex sets

TheoremFixed point theorem for maps of compact convex sets#●ready to prove

Let \(n \ge 0\) and let \(B\) be the closed unit ball about the origin in the Euclidean space \(\mathbb{R}^n\). Let \(f\) be a map from subsets of \(\mathbb{R}^n\) to subsets of \(\mathbb{R}^n\) that sends every nonempty compact convex subset of \(B\) to a nonempty compact convex subset of \(B\), and that is continuous in the Hausdorff metric on the nonempty compact convex subsets of \(B\). Then there is a nonempty compact convex set \(X_0 \subseteq B\) with \(f(X_0) = X_0\).

This answers Ulam's Problem 42 of the Scottish Book, which asks this for closed convex sets inside a ball \(K\); inside a ball the closed convex sets are the compact ones, and the typescript's "sphere \(K\)" is read as the solid ball. All sets are required to be nonempty, because the Hausdorff distance used is degenerate at the empty set. The answer is yes: for \(n \ge 2\) the space of nonempty compact convex subsets of \(B\) is homeomorphic to the Hilbert cube (Nadler, Quinn and Stavrokas), which has the fixed-point property; for \(n \le 1\) that space is a point or a \(2\)-cell, and Brouwer's theorem applies.

theorem_wanted fixed_point_for_convex_hyperspace_map :
    ∀ (n : ℕ) (f : Set (EuclideanSpace ℝ (Fin n)) → Set (EuclideanSpace ℝ (Fin n))),
      (∀ X : Set (EuclideanSpace ℝ (Fin n)),
        X ⊆ Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1 → IsCompact X →
        Convex ℝ X → X.Nonempty →
        f X ⊆ Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1 ∧
        IsCompact (f X) ∧ Convex ℝ (f X) ∧ (f X).Nonempty) →
      HausdorffContinuousOnConvexCompacts f
        (Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1) →
      ∃ X₀ : Set (EuclideanSpace ℝ (Fin n)),
        X₀ ⊆ Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1 ∧ IsCompact X₀ ∧
        Convex ℝ X₀ ∧ X₀.Nonempty ∧ f X₀ = X₀

Sources

• S. Ulam, Problem 42 of the Scottish Book; S. B. Nadler, J. Quinn and N. Stavrokas, Pacific J. Math. 83 (1979), 441-462; the fixed-point theorem for the Hilbert cube; see the commentary in R. D. Mauldin (ed.), The Scottish Book, 2nd ed. (Birkhäuser, 2015).

Uses 1

Statement usesHausdorff continuity on compact convex sets

1 fully proved1 ready to prove

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