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The hyperplane conjecture for convex bodies

A convex body K⊂RnK\subset\mathbb{R}^n is a compact convex set with nonempty interior; its barycenter is the centroid with respect to Lebesgue measure, and Vol⁡\operatorname{Vol} denotes Lebesgue volume. The notation θ⊥\theta^\perp denotes the hyperplane orthogonal to a direction θ\theta.

Hyperplane conjecture. There exists a constant C>0C>0 such that for every nn and every convex body K⊂RnK\subset\mathbb{R}^n of volume 11 and barycenter at the origin, there is a direction θ\theta such that

Vol⁡(K∩θ⊥)≥C.\operatorname{Vol}(K\cap\theta^\perp)\ge C.

This is also known as the slicing problem and asks for a dimension-independent lower bound on a central hyperplane section of every normalized convex body. The source presents it as a famous conjecture; no resolution is supplied here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The hyperplane conjecture for convex bodies

    Let n∈Nn\in\mathbb{N}, let K⊂RnK\subset\mathbb{R}^{n} be a convex set of unit volume, and let HH be a hyperplane in Rn\mathbb{R}^{n}. Hyperplane conjecture. There exists a universal positive constant cc, independent of nn, such that one can choose HH with the (n−1)(n-1)-dimensional volume of K∩HK\cap H at least cc. This is a longstanding open problem in convex geometry, also known as the slicing problem; it has several equivalent geometric and functional-analytic formulations.

    source: Meik Dörpinghaus, “A Lower Bound on the Entropy Rate for a Large Class of Stationary Processes and its Relation to the Hyperplane Conjecture”, arXiv:1512.05423 (2017).

Sources & referencesView supporting material

Primary source

Olivier Guédon, “Concentration phenomena in high dimensional geometry”, arXiv:1310.1204 (2013).

convex-geometrygeometric-measure-theoryslicing-problem

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