Convex and discrete geometry is one of the most intuitive subjects
in mathematics. One can explain many of its problems, even the most
difficult - such as the sphere-packing problem (what is the densest
possible arrangement of spheres in an n-dimensional space?) and the
Borsuk problem (is it possible to partition any bounded set in an
n-dimensional space into n+1 subsets, each of which is strictly
smaller in "extent" than the full set?) - in terms that a layman
can understand; and one can reasonably make conjectures about their
solutions with little training in mathematics.
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