Sheared Cube Polar

The polar of D = {±v₁, ±v₂} in the plane, as in Example 3.1 and Figure 2. Its vertices are ±v₁* ± v₂*, where v₁*, v₂* is the dual basis. Draw random vertices and watch the mean of ‖x‖² converge to ‖v₁*‖² + ‖v₂*‖² ≥ 2.

unit circle in gray · dashed circle through the farthest vertex

Angle between v₁ and v₂

angle

v₁ = e₁ is fixed; v₂ rotates. In the coordinates ξᵢ = ⟨x, vᵢ⟩ the polar is always the square [−1, 1]².

The averaging argument

‖v₁*‖
‖v₂*‖
E‖x‖² = ‖v₁*‖² + ‖v₂*‖²
lower bound n: 2
max ‖x‖²
cm(D) = 1 ⁄ max ‖x‖
bound 1 ⁄ √n: 0.707

draws: 0
last vertex (ξ₁, ξ₂): —
its ‖x‖²: —
running mean of ‖x‖²: —

Show

dual basis v₁*, v₂*

vertex labels (ξ₁, ξ₂)

circle through the farthest vertex

Legend

the polar P(D), sides ξᵢ = ±1 dual basis: vᵢ* ends on the side ξᵢ = 1, so ‖vᵢ*‖ ≥ 1 last random vertex drawn