It seems that the shape of the outer one is not restricted to squares, but can also be rectangles?

echo

I think the outer one could be any rectangle since we can generate uniform samples for rectangles easily. This square is just the smallest rectangle that encloses the circle.

nrauen

I understand it is possible to do a more explicit version of a sphere, but is there an equivalent rejection sampling method for higher dimensions?

silentQ

@nrauen Sure, you can just generate a random point in some cube (or rectangular bounding box), and then check if the point is within the shape you're interested in. For example, for a sphere, you could check if it is < r away from the sphere's center. But as you move to higher and higher dimensions, your probability of success decreases.

It seems that the shape of the outer one is not restricted to squares, but can also be rectangles?

I think the outer one could be any rectangle since we can generate uniform samples for rectangles easily. This square is just the smallest rectangle that encloses the circle.

I understand it is possible to do a more explicit version of a sphere, but is there an equivalent rejection sampling method for higher dimensions?

@nrauen Sure, you can just generate a random point in some cube (or rectangular bounding box), and then check if the point is within the shape you're interested in. For example, for a sphere, you could check if it is < r away from the sphere's center. But as you move to higher and higher dimensions, your probability of success decreases.