 
Summary: Pal's problem for continuous curves
Konstantin Athanassopoulos
Department of Mathematics, University of Crete, GR71409 Iraklion, Greece
In 1940 J.F. Pal posed the following problem [2]: Let (X, ) be a metric space and
let c : [0, 1] X be a continuous curve and n a positive integer. Is there a partition
0 < s1 < s2 < · · · sn < 1 such that (c(si1), c(si)) = (c(si), c(si+1)), for i = 1, 2,..., n,
where s0 = 0 and sn+1 = 1? One can generalize the problem by taking X to be merely
a topological space and : X × X R+ a continuous function such that (x, y) = 0
if and only if x = y (not necessarily a distance function). Moreover, one can require
i1(c(si1), c(si)) = i(c(si), c(si+1)), for i = 1, 2,..., n, where 0 > 0, 1 > 0,...,
n > 0 are given real numbers.
Of course the case n = 1 is trivial, because if F : [0, 1] R is the continuous function
F(s) = 1(c(s), c(1))  0(c(0), c(s)),
then, if c(0) = c(1), we have F(0) > 0 and F(1) < 0 and by the Intermediate Value
Theorem there exists some 0 < s1 < 1 such that F(s1) = 0. Every point in F1(0) is a
solution to the problem.
In 1954 K. Urbanik gave a proof of the existence of a solution to Pal's problem based
on the Brouwer fixed point theorem [6]. Another proof based again on the Brouwer fixed
point theorem was presented recently in [4]. We present a new proof of the existence
of solution to Pal's problem. The core of the argument is a generalization of the above
