Suppose you must guess the value of an unknown number x in the interval [a, b] belongs (0,Inf) and suppose you will be forced to pay a fine based on the relative error of your guess. How should you guess if you want to minimize the worst fine that you would have to pay?

I simulated stuff to get the answer. I knew all I had to check Arithmetic
Geometric and Harmonic mean

> a <- 2
> b <- 12
> N <- 1e+05
> x <- runif(N, a, b)
> AM <- (a + b)/2
> GM <- sqrt(a * b)
> HM <- 2 * ((1/a + 1/b)^-1)
> test1 <- function(x) {
+     abs(AM - x)/x
+ }
> test2 <- function(x) {
+     abs(GM - x)/x
+ }
> test3 <- function(x) {
+     abs(HM - x)/x
+ }
> t1 <- sapply(x, test1)
> t2 <- sapply(x, test2)
> t3 <- sapply(x, test3)
> boxplot(t1, t2, t3)

MinimaxCharacterization-001.jpg

The illustration shows that it is safe to go with HM as the variance is less and there are no outliers.