The convex hull of a set of points can be defined as the set of all convex combinations of the points in the set. For example and for contrast with my question, in the following two-dimensional diagram, the shape defined by the outermost lines is the convex hull of $A \cup B$.


I have two related questions. First, is there a name for the set of all convex combinations of a point from $A$ and a point from $B$ (which is the set bordered in heavy red lines in the diagram)?

Second, does there exist a well-known operation which preserves even more concavities, while still joining two arbitrary shapes (connected closed sets of points)? For example, if $A$ and $B$ had notches that were facing more upward or downward, or if the existing notches were less conveniently shaped (e.g. consider the pair of shapes "ↄc"), or if there were interior holes, then this operation would ideally preserve those concavities as well as the ones straightforwardly facing away from the other shape.

A sketch of one possible such operation follows. Let $\text{Conv}(X)$ be the convex hull of $X$.

  1. Let $A' = \text{Conv}(A) - A$, and $B' = \text{Conv}(B) - B$, the points added by the hulls for each shape (blue-bordered triangles in the diagram).
  2. Let $C = \text{Conv}(A \cup B) - \text{Conv}(A) - \text{Conv}(B)$, the points added only by the hull of both shapes (the quadrilateral in the diagram).
  3. The result is the union of $A$, $B$, $C$, and every connected component of $A' \cup B'$ which is adjacent to $C$. ("Adjacent to" needing some formalization to be precise with regard to open vs. closed sets; if we were dealing strictly in polygons I would say "shares an edge with".)

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