2
$\begingroup$

Given two convex polytopes defined as $x \in \mathcal{A}$ iff $Ax \leq c$ and $x \in \mathcal{B}$ iff $Bx \leq d$, does anyone know about (if possible easy to test) conditions of $\mathcal{A}$ and $\mathcal{B}$ sharing sharing edges, vertices or facets? Can we also consider cases when the edge of one polytope is on the face of the other, crossing its vertices?

**Correction: What I am really interested is if it possible to determine (analytically) if all the vertices of $\mathcal{A} \cap \mathcal{B}$ are also vertices of $\mathcal{A}$, however not necessarily vertices of $\mathcal{B}$.

Is it possible to say more about the special case when $A \in \{-1, 0, 1\}^{n\times n},\; B \in \{-1, 0, 1\}^{n\times n},\; c\in \{0, 1\}^{n},\; d\in \{-1, 0, 1\}^{n}$?

  • 1
    Hint: Look at the intersection of both polyhedra and compute its dimension. Now compare this dimension to the dimension of $\mathcal A$ and $\mathcal B$. – 2017-01-26
  • 1
    2nd hint: To compute the dimension have a look at "8.3 Computing H-dimension" on page 61 of https://www.inf.ethz.ch/personal/fukudak/lect/pclect/notes2016/PolyComp2016.pdf. – 2017-01-26
  • 0
    @WillemHagemann: Thanks for the hints, actually they made me realized that the question does not exactly says what I would like to have. See the correction. – 2017-01-27
  • 0
    Thanks for clarification of your question. A naive approach would be to enumerate all vertices of $\mathcal A$ and $\mathcal A \cap \mathcal B$. But this approach is rather inefficent. Perhaps you could use a variant of the backtrack algorithm stated in "Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron" (Fukuda, Liebling & Margot). A parallel enumeration of the faces of $\mathcal A$ and $\mathcal A \cap \mathcal B$ would provide many insights how the different faces relate to each other. Unfortunately, I've got no better idea yet. – 2017-01-30

0 Answers 0