# Euclidean geometry – covering the disk with a family of infinite dimensions – the convex continuation

To let $$(U_n) _n$$ be an arbitrary sequence of open convex subsets of the unit disk $$D (0,1) subseteq mathbb {R} ^ 2$$ s.t. $$sum_ {n = 0} ^ infty lambda (U_n) = infty$$ (Where $$lambda$$ is the Lebesgue measure). Is there a sequence? $$(q_n) _n$$ in the $$mathbb {R} ^ 2$$ s.t. $$D (0,1) subseteq bigcup_ {n = 0} ^ infty (q_n + U_n)$$?

With the notation $$q_n + U_n$$, I mean
$$q_n + U_n: = {x in mathbb {R} ^ 2 | x-q_n in U_n }$$

This question is very similar to this one, but I was asked in the comments to ask anyway.