  We construct log canonical pairs $(X,B)$ with $B$ a nonzero reduced divisor
and $K_X+B$ ample that have the smallest known volume. We conjecture that our
examples have the smallest volume in each dimension. The conjecture is true in
dimension 2, by Liu and Shokurov. The examples are weighted projective
hypersurfaces that are not quasi-smooth.
  We also develop an example for a related extremal problem. Esser constructed
a klt Calabi-Yau variety which conjecturally has the smallest mld in each
dimension (for example, mld $1/13$ in dimension 2 and $1/311$ in dimension 3).
However, the example was only worked out completely in dimensions at most 18.
We now prove the desired properties of Esser's example in all dimensions (in
particular, determining its mld).
