How to prove that the erosion of a convex set by a convex set will result in a convex set?
(Convexity is closed under intersection) Given a family of convex set $C_i$, where $i\in I$ for some index set $I$, then $\cap_{i\in I}C_i$ is convex.

In other words, the intersection of convex sets are convex.

Proof. Suppose two points $p$ and $q$ are in $\cap_{i\in I}C_i$. That is, for each $i\in I$, $p\in C_i$ and $q\in C_i$. Consider $L$, the line segment between $p$ and $q$. By the definition of convexity, $L\subseteq C_i$. That is $L\subseteq \cap_{i\in I}C_i$. By the definition of convexity again, $\cap_{i\in I}C_i$$ is convex.

(Convexity is closed under translation) If $C$ is a convex set, then $C_t=\{x+t\mid x\in C\}$ is also convex.

In other words, if you translate a convex set, you will get a convex set.

Proof is skipped.

(Convexity is closed under erosion) Suppose $A$ is a convex set. Let $B$ be another set. Then the erosion of $A$ by $B$ is convex.

Proof. I will let you figure out given the above two lemmas. Another hint is given in the statement itself, which is, you do not need $B$ to be convex at all.



Is the erosion of $A$ by $B$ always empty?

Imagine both $A$ and $B$ are disks centered at the origin. $A$ is a huge while $B$ is a very small. Then the "erosion" by $B$ can only remove a small stripe along the circumference of $A$. The inner part of $A$ will not be "eroded". So the erosion of $A$ by $B$ will keep a significant part of $A$.

You can also check an example on Wikipedia.