On Reflection Properties of Convex Regions

It is well known that any ray of light passing thru a focus of an ellipse will pass thru the other focus after a single reflection from the ellipse boundary. If A and B are the foci of an ellipse, this property of rays holds both ways (those passing thru A meet at B and vice versa). Two basic queries on this reflection property:

1. Is there a closed convex region C with the property: there exists a pair of points A and B within C such that all rays thru A will reflect once on C and pass thru B but not all rays thru B will pass thru A after one reflection from C?

2. Is there a closed convex region C such that: there is a pair of points A and B in the interior such that all rays thru A pass thru B after exactly 2 reflections from C? Note: This question can have 'one-way' (convergence only of rays thru A at B) and 'both-ways' variants

Update (June 28th, 2020):
See this page for a nice discussion on the above questions:
https://mathoverflow.net/questions/364134/on-reflection-properties-of-convex-regions