Main

## 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.