Re: Variation of a centroid

On Mon, 10 Apr 2006 02:22:43 -0400, "Stephen J. Herschkorn"
<sjherschko@xxxxxxxxxxxx> wrote:

Let f be a positive, continuous function on the reals such that f(-x)
f(x) for all positive x. For nonnegative t, let a(t) be the
abscissa of the centroid of the region {(x,y) in R^2: |x| <= t, 0 <=
y <= f(x)}. It is intuitive that a is strictly decreasing. Is the
formal proof of this obvious? (A sincere question.) If so, what is
this proof?

I think the claim is false.

A counterexample can be created as follows ...

Let f have a central piece for which the centroid is strongly biased
left. Then define f on the left and right of that central piece to be
almost symmetrical (hence almost unbiased). Thus, as a function of t,
the outer piece will force the x-value of the centroid to the right,
back towards 0. Hence for the values of t corresponding to the outside
pieces, the x-value of the centroid will be an increasing function of
t.

Here's an actual counterexample based on the above idea ...

Define f as a piecewise function as follows:

f(x) =
5 if x <= -1
3 - 2*x if -1 < x <= 1
9*x - 8 if 1 < x <= 4/3
4 if x > 4/3

Then a(t) is increasing on the interval [1,2].

quasi
.

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