Geometry (Convex Polygons)

Question

Let P be a set of points in the plane. Let P1 be the convex polygon whose vertices are points from P and that contains all points in P. Prove that this polygon P1 is uniquely defined, and that it is the intersection of all convex sets containg P

Answer 1

Your statement is not true.

First, if the set $P$ is unbounded, the minimal enclosing convex set will not be a "polygon" in the usual sense, since some "edges" go off to infinity. (Example: $P$ is the x-axis.) So you either need to add the condition that $P$ is bounded or modify the definition of "polygon."

Second, if $P$ contains infinitely many points, the "polygon" may have curved sides. (Example: $P$ is the unit circle.) You can avoid this by requiring $P$ to be finite or again modifying your definition of "polygon."

Third, if the points in $P$ are collinear, the set $P_1$ will be a line or line segment, again not a polygon in the usual sense. (Example: $P$ is the x-axis, or just the origin.)

You can rescue your statement by requiring $P$ to be a finite set and by allowing degenerate "polygons" like line segments and points to be polygons. There are also other possible modifications. Let me know if you need a proof of a appropriately modified statement.

(Note: I was given a very similar problem in an application to attend a course at Ohio State University when I was a high school student. I got one proof but could not complete an attempted alternate approach--until three years later when I had an "aha" moment when I was walking to school one morning.)