Vertex Loci of Max-Indicator Polytopes and Vertex Retrieval in Ball Polyhedra
The so called "max-indicator polytopes" were introduced in \cite{mainB} as an intersection of half-spaces related to the problem of finding the farthest in an intersection of balls $\mathcal{Q}$, from a given target point $C_S$. It was shown that if the target point belongs to the convex hull of the balls centers, then a specific intersection of half-spaces forms a single real variable parameterized polytope family which is used to characterized the extreme points. This paper, under the assumptions that the intersecting balls centers lie on a sphere and all have equal radii with only one vertex $x$, of $\mathcal{Q}$, meeting $1_{n \times 1}\cdot x = 0$, we provide an efficient algorithm to retrieve that vertex. The main idea behind the result is that the max-indicator polytopes vertices, as the family parameter varies, slide on some lines, that we call vertex locii. These lines pass through the space origin and the vertices of $\mathcal{Q}$. Perturbing the target point $C_S$ up and down along the direction $1_{n\times 1}$ we are able to show that the intersection of two simetrically perturbed max-indicator polytopes is strictly included in the unperturbed polytope, with the only boundary common points being on the line connecting the origin with the vertex of $\mathcal{Q}$ meeting $1_{n\times 1}^T\cdot x = 0$. As such, under the said assumptions, the only such vertex can be retrieved using linear programming.