On the diminishing process of B. Tóth

Original · EN

Let K and K₀ be convex bodies in Rᵈ, such that K contains the origin, and define the process (Kₙ, pₙ), n ≥ 0, as follows: let pₙ₊₁ be a uniform random point in Kₙ, and set Kₙ₊₁ = Kₙ ∩ (pₙ₊₁ + K). Clearly, (Kₙ) is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in Rᵈ. We study this process for K being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.