What exactly does a compact convex set mean.?

Let $A$ be a compact convex set in $\mathbb{R}^2$.

What does ''compact convex set'' mean?

What I understand: We have a "bunch" of real points $(x,y)$ in the plane. Any two of them satisfies the fact that a line drawn between them is fully inside this "bunch" of points.

So can this "bunch" be a polygon? (I think it can.)
Also, how does the compact part fit in and what does it mean?
How do I make one in the plane?
Also what is a "compact convex polygon" if it is possible?

Solution 1:

A convex set is one where the line between any two points in the set lies completely in the set. Polygons can be convex or concave - for example, a crescent moon is a concave set and you can approximate it with a polygon that stays concave.

A compact set, at least in the real Cartesian plane, is one that is both closed and bounded, which roughly means that (a) it includes its own boundary and (b) it has no points going to infinity. Polygons, assuming you include both the boundary and interior, are compact in $\mathbb{R}^2$.

So yes, a polygon may be both convex and compact, but not every convex compact set is a polygon, and not every polygon is a convex compact set.

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