  We prove that the Fibonacci word $f$ satisfies among all characteristic
Sturmian words, three interesting extremal properties. The first concerns the
length and the second the minimal period of its palindromic prefixes. Each of
these two properties characterizes $f$ up to a renaming of its letters. A third
property concerns the number of occurrences of the letter $b$ in its
palindromic prefixes. It characterizes uniquely $f$ among all characteristic
Sturmian words having the prefix $abaa$.
