Thanks for amazing lecture. I have a silly question. A polytope can contain a small Euclidean ball implying that ball also has a small covering number as polytope? How to understand this?
The ball will have a radius that will decay with dimension. It will decay like about 1/sqrt(ambient dimension), I think. This small radius should allow the small covering number of the ball to make sense.
It is also important to realize that the result is only polynomial in m for fixed epsilon. For fixed epsilon and high enough dimension, the inner ball will be small enough to be covered by a single epsilon-ball.
Thanks to the the guys who asked questions!
Thanks for amazing lecture. I have a silly question. A polytope can contain a small Euclidean ball implying that ball also has a small covering number as polytope? How to understand this?
The ball will have a radius that will decay with dimension. It will decay like about 1/sqrt(ambient dimension), I think. This small radius should allow the small covering number of the ball to make sense.
It is also important to realize that the result is only polynomial in m for fixed epsilon. For fixed epsilon and high enough dimension, the inner ball will be small enough to be covered by a single epsilon-ball.