Hyperplane slicing problem

DefinitionConvex bodies in \(\mathbb{R}^n\)#✓fully proved

A set \(K\) in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) is a convex body when it is convex, compact and has nonempty interior.

def IsConvexBody (n : ℕ) (K : Set (EuclideanSpace ℝ (Fin n))) : Prop :=
  Convex ℝ K ∧ IsCompact K ∧ Set.Nonempty (interior K)

Sources

• OWR-12337-001 (DOI 10.4171/owr/2013/09): "convex body K ⊆ ℝⁿ".

Used by 1

Used byHyperplane slicing conjecture

DefinitionHyperplanes in \(\mathbb{R}^n\)#✓fully proved

A set \(H\) in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) is a hyperplane when \(H = \{x \in \mathbb{R}^n : f(x) = t\}\) for some nonzero linear functional \(f \colon \mathbb{R}^n \to \mathbb{R}\) and some \(t \in \mathbb{R}\).

def IsHyperplane (n : ℕ) (H : Set (EuclideanSpace ℝ (Fin n))) : Prop :=
  ∃ (f : LinearMap (RingHom.id ℝ) (EuclideanSpace ℝ (Fin n)) ℝ) (t : ℝ), f ≠ 0 ∧
    H = {x : EuclideanSpace ℝ (Fin n) | f x = t}

Sources

• OWR-12337-001 (DOI 10.4171/owr/2013/09): "hyperplane section".

Used by 1

Used byHyperplane slicing conjecture

DefinitionHyperplane slicing conjecture#✓fully proved

The statement that there is a constant \(c > 0\) such that for every \(n \ge 1\) and every convex body \(K \subseteq \mathbb{R}^n\) (a convex compact set with nonempty interior) of Lebesgue volume \(1\) there exist a hyperplane \(H \subseteq \mathbb{R}^n\) (a set \(\{x : f(x) = t\}\) for a nonzero linear functional \(f\) on \(\mathbb{R}^n\) and some \(t \in \mathbb{R}\)) and an isometry \(\varphi \colon \mathbb{R}^{n-1} \to \mathbb{R}^n\) with image \(H\) such that the \((n-1)\)-dimensional Lebesgue volume of \(\varphi^{-1}(K \cap H)\) is at least \(c\).

def conjecture : Prop :=
  ∃ (c : ℝ), 0 < c ∧
    ∀ (n : ℕ), 1 ≤ n →
      ∀ (K : Set (EuclideanSpace ℝ (Fin n))),
        IsConvexBody n K →
          MeasureTheory.volume K = 1 →
            ∃ (H : Set (EuclideanSpace ℝ (Fin n))),
              IsHyperplane n H ∧
                ∃ (φ : EuclideanSpace ℝ (Fin (n - 1)) → EuclideanSpace ℝ (Fin n)),
                  Isometry φ ∧ Set.range φ = H ∧
                    ENNReal.ofReal c ≤ MeasureTheory.volume (Set.preimage φ (K ∩ H))

Sources

• OWR-12337-001 (DOI 10.4171/owr/2013/09), where it is posed as an open problem; it has since been resolved.

Uses 2 · Used by 1

Statement usesConvex bodies in \(\mathbb{R}^n\) · Hyperplanes in \(\mathbb{R}^n\)

Used byHyperplane slicing theorem

TheoremHyperplane slicing theorem#●ready to prove

There is a constant \(c > 0\) such that for every \(n \ge 1\) and every convex body \(K \subseteq \mathbb{R}^n\) (a convex compact set with nonempty interior) of Lebesgue volume \(1\) there exist a hyperplane \(H \subseteq \mathbb{R}^n\) (a set \(\{x : f(x) = t\}\) for a nonzero linear functional \(f\) on \(\mathbb{R}^n\) and some \(t \in \mathbb{R}\)) and an isometry \(\varphi \colon \mathbb{R}^{n-1} \to \mathbb{R}^n\) with image \(H\) such that the \((n-1)\)-dimensional Lebesgue volume of \(\varphi^{-1}(K \cap H)\) is at least \(c\).

This is Bourgain's slicing conjecture, proved by Klartag and Lehec building on Guan's bound (Theorem 1.1); equivalently, the isotropic constant of convex bodies is bounded by a universal constant.

public theorem_wanted conjecture_holds : conjecture

Sources

• B. Klartag, J. Lehec, Affirmative resolution of Bourgain's slicing problem using Guan's bound, Geometric and Functional Analysis (2025), arXiv:2412.15044, https://arxiv.org/abs/2412.15044.

Uses 1

Statement usesHyperplane slicing conjecture

3 fully proved1 ready to prove

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