Main

Special Case 1

Elliptope: E_3

E_3 = \left\{[x,y,z] \in \mathbb{R}^{3}\,\Big|\, \begin{bmatrix} 1 & x& y\\x & 1& z\\y & z& 1 \end{bmatrix} \succeq 0 \right\}.

The elliptope E_3 is well understood. It is a 3-dimensional spectrahedron, also known as the inflated tetrahedron.

Algebraic boundary

The algebraic boundary is given implicitly by the following ternary cubic (known as Cayley cubic):

\det(\begin{bmatrix} 1 & x& y\\x & 1& z\\y & z& 1 \end{bmatrix} = 1 + 2xyz-x^2-y^2-z^2=0.

Dual body

The dual body is defined as

E_3^* = \left\{l \in (\mathbb{R}^{3})^*\,\Big|\, l(x,y,z) < 1 \,\forall\, [x,y,z] \in E_3\right\}.

it can be described as the spectrahedral shadow (aka projection of a spectrahedron)

E_3^* = \left\{ [u(M), v(M), w(M)]^T \in \mathbb{R}^{3} \,\Big|\, M\bullet I_3 < 1, M\succeq 0 \right\},

were

u(M) = - \begin{bmatrix}0 &1 &0\\ 1& 0& 0\\ 0& 0& 0\end{bmatrix} \bullet M,
v(M) = - \begin{bmatrix}0 &0 &1\\ 0& 0& 0\\ 1& 0& 0\end{bmatrix} \bullet M

and

z(M) = - \begin{bmatrix}0 &0 &0\\ 0& 0& 1\\ 0& 1& 0\end{bmatrix} \bullet M.

The algebraic boundary of the dual body is given by the following quartic surface (known as Steiner surface):

x^2y^2+x^2z^2+y^2z^2-2xyz = 0

Figure


Fig. 1: Cayleys cubic surface, the elliptope E_3 is the yellow part of it.

Fig. 2: Steiner surface, the algebraic closure of the dual to the elliptope E_3.

Faces

  • 4 x 2-dimensional protrusions of extreme points
  • 6 edges
  • 4 vertices

Protrusion lattice

Vertices

The elliptope has four vertices, corresponding to the following rank one matrices:

X_1 = \begin{bmatrix}1 & 1& 1\\1 & 1& 1\\1 & 1& 1 \end{bmatrix}, X_2 = \begin{bmatrix}1 & -1& -1\\-1 & 1& 1\\-1 & 1& 1 \end{bmatrix}, X_3 = \begin{bmatrix}1 & 1& -1\\1 & 1& -1\\-1 & -1& 1 \end{bmatrix}, X_4 = \begin{bmatrix}1 & -1& 1\\-1 & 1& -1\\1 & -1& 1 \end{bmatrix}

Edges

One edge between each pair of two vertices. The corresponding matrices are rank two and the Hessian of the algebraic boundary is rank deficient at these points.

Two dimensional protrusions

Four 2-dimensional protrusions of extreme points, each of corresponding to an inflated facet of a tetrahedron. The four different protrusions correspond to rank two matrices and can be parametrized explicitly by the following Gram matrix:

X^{\rm 2D} = \begin{bmatrix}1 & \langle v_1, v_2\rangle& \langle v_1, v_3\rangle\\\langle v_2, v_1\rangle & 1& \langle v_2, v_3\rangle\\\langle v_3, v_1\rangle & \langle v_3, v_2\rangle& 1 \end{bmatrix},

where v_1 = [1,0]^T, v_2 = [\cos(\alpha),\sin(\alpha)]^T, v_3 = [\cos(\beta),\sin(\beta)]^T. The range of the two angles \alpha,\beta determines which protrusion is parametrized:

  1. 0<\alpha < \beta<\pi
    closure contains X_1,X_2 and X_3
  2. 0< \beta < \alpha <\pi
    closure contains X_1,X_2 and X_4
  3. 0< \alpha < \beta-\pi <\pi
    containing X_1,X_3 and X_4
  4. 0< \beta-\pi < \alpha < \pi
    closure X_2,X_3 and X_4

The following picture illustrates the different protrusions:


Fig. 3: Different protrusions of the elliptope E3.