3. Intersections with State-space Partitions

For every k \in \mathbf{K} and its associated simplicial partition element Q_{k} \subset D with positive volume, the set-valued image P(a)(Q_{k}):

  1. is another N_{z}-simplex contained in the unit N_{z}-simplex D; and

  2. intersects with:

    • at least one partition element Q_{k'} where k' \in \mathbf{K} and
    • at most all partition elements Q_{1}, ..., Q_{K};

3.1. Polytope intersection problems

Denote

\mathbf{I}(a,k) := \left\{ k' \in \mathbf{K} : P(a)(Q_{k}) \cap Q_{k'} \neq \emptyset \right\}, \qquad \forall a \in A, k \in \mathbf{K},

as the sets of indexes to respective partition-elements—i.e. k' \mapsto Q_{k'}—that contain non-empty intersections with each simplicial image P(a)(Q_{k}). Each nonempty intersection, induced by each (a,k) \in A \times \mathbf{K} and P(a), is described by

Poly_{k'(a,k)} := \left\{ \lambda' \in D : \lambda' \in P(a)(Q_{k}) \cap Q_{k'} \neq \emptyset \text{ and } k' \in \mathbf{I}(a,k) \right\}.

Note

Each intersection Poly_{k'}, for each k \in \mathbf{K} and each a \in A, is a polytope, and is at least a simplex, and is a subset of partition element Q_{k'}, where k' := k'(a,k).

These nonempty intersections are such that

\bigcup_{k' \in \mathbf{I}(a,k)} Poly_{k'} :=: P(a)(Q_{k}).

Example

If N_{z} = 3, then D is a unit 2-simplex, and each Poly_{k'} can be a polygon or a triangular subset in D.