Welcome to another hypothetical art gallery tour. Today's tour will feature convexity.

A figure has ** convexity** if all of its interior angles are less than \(180^\circ\). This way of thinking about things focuses our attention on the edges of the figure rather than the interior. What if we think about convexity in terms of line of sight instead?

In yesterday's problem we tried to place a guard in an art gallery inside a polygon so that every part of the interior could be "seen." We saw that three different concave (i.e. **not** convex) shapes can have different solutions. While **A**, **B**, and **C** are all concave, two of them **can** be guarded by a single person, but it's impossible for any one guard to see all of **B** from inside.

It turns out if we want art gallery problems to be interesting, they *must* be made with concave figures, because **every convex polygon can be guarded by a single guard standing anywhere.** If you're not convinced, consider that the inside of a convex gallery *never* has a corner that can block your view.

Saying that convex galleries only need one guard is just another way of stating a fundamental property of convex polygons: any vertex can be connected with any other without passing through one of the sides. This property is always true for convex polygons and it's *always false* for concave polygons (see the image above for an example).

This property is important for art gallery problems because a guard's line of sight is constrained by all the lines between the guard's position and the vertices of the gallery.

Some concave galleries, like **A** and **C** above, only need a single guard. Others, like **B**, need more. What makes these shapes different? Today's problem is a first step toward classifying galleries. Does the condition below *always* put a concave gallery in the "one-guard" category?