What is the name for this subset of concave shapes?

Question

So for example, using only pixels to form the shapes, though that isn't required to meet the criteria (assume they are joined when on adjacent rows).

Meet criteria:

  ██
███
███

██
██
██████

 ███
████████
  ███

Do not meet criteria:

█ █
███

████████
███ ████

  █████
 ██   ██
  ██ ██

The criteria for being part of the subset of concave shapes, are

a) Concave

b) If you have a rectangle with the same orientation as the shape you can test if each corner of the rectangle is within the shape, and if they all are then you can conclude that the rectangle is fully "within" the shape, as in, no part of the rectangle is outside.

For example with b, you could show the final shape doesn't meet the criteria if the rectangle was such that each corner was just outside the "hole" in the centre.

---

Someone reasonably pointed out that the rectangle cannot have an orientation the same as the shape if the shape isn't made of pixels, i.e. all straight edges are in one direction or perpendicular to that direction. This would be a criterion:

c) All edges are straight and in one direction or perpendicular to that direction.

Answer 1

According to Rahul in this comment:

I believe what you are looking for is an orthogonally convex set, or more specifically, a nonconvex but orthogonally convex set.