1. Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions?

2. Can projection of $n$ dimensional convex sets in $\mathbb R^{n'}$ onto $m$ dimensional convex sets in $\mathbb R^{m'}$ with $m<m'<n<n'$ produce non-convex connected sets of dimension smaller than $m$?