An Estimate for the Hausdorff Distance between a Set and Its Convex Hull in Euclidean Spaces of Small Dimension

A Simple Closed Curve in ℝ³ Whose Convex Hull Equals the Half-Sum of the Curve with Itself

“…It is well known [3] that the Cantor set C equals the set of those numbers in [0, 1] that have only 0's and 2's in their ternary expansion; that is, 1],…”

“…Since Γ in this question is a homeomorphic image of the unit circle S 1 , it follows that Γ + Γ is a continuous image of the torus S 1 × S 1 . Therefore in smooth cases 1 2 (Γ + Γ) looks like a two-dimensional manifold with self-intersections, so it should not contain a tetrahedron. Indeed, we show that even in the rectifiable case the answer to the above question is negative:…”

“…Recall that the Minkowski sum A + B of two sets in the n-dimensional Euclidean space R n is the set {x+y ∶ x ∈ A, y ∈ B}; recall also that rA ∶= {rx ∶ x ∈ A} for r ∈ R. It follows from the Shapley Folkman theorem [1] that the sequence Γ, 1 2 (Γ + Γ), 1 3 (Γ + Γ + Γ), 1 4 (Γ + Γ + Γ + Γ), . .…”

“…If Γ is the range of a simple closed curve that bounds a convex set in R 2 , then already the second term in this sequence reaches the limit, so that 1 2 (Γ + Γ) = co(Γ) for such Γ. Recently V.N.…”

“…Question. Is there a simple closed curve in R 3 whose range Γ does not lie in any plane and such that 1 2 (Γ + Γ) = co(Γ)? Note that Γ does not lie in any plane if and only if co(Γ) contains a tetrahedron.…”