  One of the oldest and richest problems from continuous location science is
the famous Fermat-Torricelli problem, asking for the unique point in Euclidean
space that has minimal distance sum to n given (non-collinear) points. Many
natural and interesting generalizations of this problem were investigated,
e.g., by extending it to non-Euclidean spaces and modifying the used distance
functions, or by generalizing the configuration of participating geometric
objects. In the present paper, we extend the Fermat-Torricelli problem in a
two-fold way: more general than for normed spaces, the unit balls of our spaces
are compact convex sets having the origin as interior point (but without
symmetry condition), and the n given objects can be general convex sets
(instead of points). We combine these two viewpoints, and the presented
sequence of new theorems follows in a comparing sense that of theorems known
for normed spaces. Some of these results holding for normed spaces carry over
to our more general setting, and others not. In addition, we present analogous
results for related questions, like, e.g., for Heron's problem. And finally we
derive a collection of results holding particularly for the Euclidean norm.
