Interactive versions of the figures of “Positive bases of size 2n − 1 with maximal cosine measure” (G. Jarry-Bolduc and M. Krishan Lal). Choose a figure below; in the three-dimensional ones, drag to rotate and scroll to zoom.
The polar P(D) = {x : ⟨x, d⟩ ≤ 1 for all d ∈ D} of three unit vectors in the plane, as in Example 2.6. The cosine measure is the reciprocal of the distance to the farthest vertex; every farthest vertex, normalized, is a cosine vector.
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.
A positive basis of ℝ³ with five vectors, in the form of Theorem 3.2: a triangle {v₁, v₂, w₀} in a plane and a pair {v₃, w₃} that may be hooked onto the shared vector v₂ by the critical vector −c₃v₂. Drag the tips of v₁, v₂, v₃ on the sphere; drag elsewhere to rotate; scroll to zoom.
The two positive bases of Example 3.3 and Figure 3: the triangle {e₁, e₂, w₀} in the plane L, together with a pair {e₃, w₃} that is either honest (w₃ = −e₃) or hooked onto e₂. Drag to rotate, scroll to zoom.
The polar P(D) of Example 3.4, D = {e₁, e₂, e₃, w₀, w₃}, with w₀ = −(a₁e₁ + a₂e₂) and w₃ = −(b₃e₃ + c₃e₂). Drag to rotate, scroll to zoom.
The triangle D₂ = {v₁, v₂, w₀} of Lemma 3.6 and its polar in the plane, as in Figure 5. The vertices p₀, p₁, p₂ are where two constraints are tight; the shared coordinate ⟨y, v₂⟩ equals 1 on the side tangent at v₂ and reaches its floor m₀ at p₂. Drawing the vertex pᵢ with probability λᵢ = (1, a₁, a₂)ᵢ⁄Λ gives a mean-zero random point with E‖y‖² ≥ 4.
The positive basis D₃,₅ of ℝ³ from Figure 6: a regular simplex t₀, t₁, t₂ of the horizontal plane together with ±e₃. Its convex hull is a triangular bipyramid and its polar is the prism T × [−1, 1]. Drag to rotate, scroll to zoom.