| # Separation of Ellipsoids We consider the following geometrical problem: find a hyperplane that separates two ellipsoids in , or determine there is none. ## Putting a sphere in a half-space We first seek a condition that determines wether the sphere of radius centered at is entirely contained in the half-space where , are given. The condition is equivalent to the fact that In view of the Cauchy-Schwartz inequality, we have Clearly, the right-hand side is attained by some vector on the unit circle, namely . Thus, the condition we seek is ## Putting an ellipsoid in a half-space Now consider the case with an ellipsoid instead of a sphere. An ellipsoid can be described as an affine transformation of the unit sphere: where is the ‘‘center’’, and is a matrix that determines the ‘‘shape’’ of the ellipsoid. The containement condition is equivalent to Using the same argument as before, we obtain the condition Note that the condition that should be on the other side of the hyperplane is readily obtained by changing in , that is: ## Ellipsoid separation Now consider two ellipsoids where are the centers, and the shape matrices, . The hyperplane separates the two ellipsoids if and only if Thus, the existence of such a separating hyperplane is equivalent to the existence of such that Exploiting the homogeneity with respect to , we can always normalize the condition so that . The SOCP allows to determine if the ellipsoids are separable: they are if and only if . In this case, an appropriate value of is
| Separating two ellipsoids by a hyperplane.