# How Should One Define the South-Western-Most Point of a Two-dimensional Figure?

Suppose we are given a (connected and compact) two-dimensional figure $X$.

How should one best define the south-western-most point (SWMP) of $X$ mathematically, so that it agrees with one's intuition?

I know this question is a bit vague, so I will include some examples to explain what I mean.

I think for any rectangle (with sides parallel to the axes), the SWMP should be the bottom-left-hand corner as indicated below:

For a rhombus, though, the SWMP isn't maybe as obvious. Here are some options:

Finally, for a non-convex shape, a notion of SWMP is counter-intuitive; maybe it would be something like this:

One possible definition might be:

Making the center of mass the origin of the Cartesian plane, we consider all possible line segments in the third quadrant contained within $X$ andemanating from origin. We select one with the longest length and call its intersection with $X$ a SWMP.

This seems to make sense for the rectangle's SWMP, but not intuitively for some of the other shapes in general. For instance, the non-convex shape's SWMP according to this definition would either be the most western point or the most southern point, which seems intuitively false.

Thoughts?

P.S. This question is motivated by the fact that the Cape of Good Hope claims to be the SWMP of Africa.

• How about simply the place in the figure where $x+y$ is minimized. This is fine for rectangles, no help at all in choosing between your rhombus examples, and not much help with your star. – kimchi lover Sep 18 '17 at 3:26
• I don't think it's possible to satisfy your request for a SWMP that "agrees with one's intuition" when you yourself admit that "a notion of SWMP is counter-intuitive". I posit that the notion of a unique SWMP does not exist -- every point on the Pareto frontier, i.e. further South than anything to its West and further West than anything to its South, has a legitimate claim to being a SWMP. – user856 Sep 18 '17 at 13:22
• That said, the Cape of Good Hope does appear to be the point which minimizes $x+y$ on the Mercator projection. – user856 Sep 18 '17 at 13:26
• What's the southern-most point of the given rectangle? Same problem. – nickgard Sep 18 '17 at 20:16

Given a (bounded plane) set $S$, to find the extremest point in direction $D$ I suggest forming the convex hull of $S$, then the intersection of $S$ with the supporting line perpendicular to $D$. If that intersection has more than one point you have to live with the ambiguity.