**PREABMLE TO MY QUESTION**

I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many kinds of rectilinear convex hulls. For the ease of people reading my question, I am providing the important definitions using an example below. Note that a rectilinear line (or line-segment) is a straight line (or line-segment) oriented parallel to either the x -axis or the y-axis. A rectilinear curve or staircase is a (finite) line consisting of rectilinear line segments. In the example (image), the Subimage(a) shows the point set in $\mathbb{R}^{2}$ for which the various r-convex hulls are plotted.

**Definition 1 (Subimage (a))**. A point set is called rectilinear convex, or *r-convex*, if for any two of its points which determine a rectilinear line segment, the line segment lies entirely in the given set. The point set $\mathbf{P}$ is r-convex.

**Definition 2 (Subimage (b))**. The rectilinear-convex hull or *r-convex hull* of a point set is the intersection of all r -convex sets that contain the given set. The *r-convex hull* of a point set is *unique*.

**Definition 3 (Subimage (c))**. The maximal r-convex hull or *mr-convex hull* of a point set is the intersection of all closed-rectilinear-half-planes (refer figure-4 of publication) that contain it. The *mr-convex hull* of a point set is *unique*.

**Definition 4 (Subimage (d))**. The connected r-convex hull or *cr-convex hull* of a point set is a smallest connected r-convex set containing the given point set. Observe that *cr-convex hulls* of a point set are *not necessarily unique*. In Subimage (d), the staircase a − 1 − b and point c are joined with one of the many staircases possible between points 1 and c. As long as the staircase joining a − 1 − b and c is within or on the box 1 − 3 − c − 2 − 1 (shown with dotted lines), the set will remain the smallest r-convex set containing point set P.

The publication also points out (on page-162) that *r-convex hull* and *mr-convex hull* of a point are consistent with the Carathéodory's theorem where as *cr-convex hulls* are not.

**MY QUESTION**
I want to propose one more type of *r-convex hull*, namely *ucr-convex hull*. I define it as :

**Definition 5** *ucr-convex hull* is the set obtained from union of all *cr-convex hulls*. See Subimage (e).

I intuitively feel that the *ucr-convex hull* has some nice properties such as uniqueness and consistency (wrt Carathéodory's theorem for regular convex sets). BUT I don't know how to methodically go about proving these properties. Any directions?

Edit 1 : I was able to prove that ucr-covex hulls do not adhere to the Caratheodory's theorem. I am looking for answers on Uniqueness property still though.

Edit 2 : After discussions with someone, I understand that the Uniqueness of ucr-convex hull is trivial as long as the *number of smallest rectilinear paths between two points can be proven to be a countable*. I dont have any background in real analysis or core maths subjects to be able to produce a formal proof to this intuitive concept. Any help would be appreciated.

rectilinear line segmentin a comment below mine? $\endgroup$ – Włodzimierz Holsztyński Sep 29 '16 at 19:17