id: 06455890
dt: j
an: 2016b.00116
au: Ziegler, Günter M.
ti: Cannons at sparrows.
so: Eur. Math. Soc. Newsl. 95, 25-31 (2015).
py: 2015
pu: European Mathematical Society (EMS) Publishing House, Zurich
la: EN
cc: A85 G95 F65
ut: dissection; perimeter; area; Voronoi diagram; configuration space; test
map; obstruction; equivariant obstruction theory; permutahedron;
Pascal’s triangle
ci:
li:
ab: This fascinating and entertaining article begins with an apparently simple
problem in convex geometry, due to R. Nandakumar and R. Ramana Rao:
given a convex shape, can it be partitioned into $N$ pieces so that all
pieces have equal area and perimeter? (This is different, except when
$N=2$, from the “cake and icing problem" of dividing the region into
pieces that each have the same area and the same portion of the
original body’s perimeter.) The $N=2$ case can be solved easily (if
unsportingly) by the use of the Borsuk-Ulam theorem. This paper
summarizes further progress on the problem, which involves even more
diverse heavy artillery. To even represent the topological space of
equal-area partitions requires ideas from the theory of {\it optimal
transport}, and particularly a 1938 theorem of Kantorovich on weighted
Voronoi diagrams. Once posed in this way, the problem can be converted
into a problem in equivariant algebraic topology via the
“configuration space / test map scheme". If there is {\it no}
solution to the original problem, then there is an equivariant map from
a certain ($n-1$)-dimensional cell complex to the ($n-2$)-sphere. The
existence or nonexistence of this map can be determined using
equivariant obstruction theory. This in turn leads, via permutahedra,
to a question (solved over a hundred years ago) about prime factors of
entries in Pascal’s triangle, and the final result of the paper, due
to the author and Blagojević: if $N$ is a prime power, the desired
dissection exists. The paper is clear and well-written, though I was
puzzled to find “equivariant obstruction theory" abbreviated as
“EOS" twice. This appears to be a typo: it’s “EOT" on two other
occasions.
rv: Robert Dawson (Halifax)