What exactly does a compact convex set mean.?

Question

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?

Answer 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.

Answer 2

A compact and convex set refers to a set of points that follow the property:

1. The set is convex, that is, any line that connects any and all the two points chosen from the set lies in the set. Think of the parabola y=x^2. Now choose any 2 points lying inside or on the parabola and try to connect them by drawing a straight line. You will observe that all such lines will lie inside the parabola, which means that the curve is convex.
2. Compactness refers to the property that the set shall be closed and bounded. The example of parabola from the previous point does not hold true here because it was not a bounded curve. A disk of finite radius for eg, would be a compact and convex curve as it is bounded, has no holes and is convex. Note: Don't confuse the disk with a circle. A circle only refers to the boundary, excluding the interior that is why it is not possible to connect any 2 points without moving the pencil out of it.