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Let $n$ be a positive integer greater than $2$. Let $X(1),X(2),\cdots,X(n)$ and $ Y(1),Y(2),\cdots,Y(n)$ be two strictly increasing sequences of n real numbers each, listed in order of increasing size. This question asked whether there always existed a polynomial $P(X)$ such that

  • $Y(m)=P(X(m))~~~~~~~~~~~~~~~~ \forall m=1,2,\cdots,n$

  • $P(X)$ is strictly increasing throughout the closed interval of real numbers whose end-points are $X(1)$ and $X(n)$.

There were numerous responses to this question which gave an affirmative answer and discussed a number of related results in this area.

I have been trying to prove whether the following additional constraints can also be imposed on the polynomial $P(X)$. Let $T$ be the triangle which is the convex hull of the points $(X(1),Y(1))$, $(X(1),Y(n))$ and $(X(n),Y(n))$. If all of the points $(X(m),Y(m))$ are in the interior of $T$ for each positive integer $m$ which is greater than $1$ and less than $n$, can the point $(X,P(X))$ always be in the interior of $T$ for each real number $X$ in the open interval whose end-points are $X(1)$ and $X(n)$?

My second question is: If the points $(X(m),Y(m))~~\forall m=1,2,\cdots,n$ are all points of $T$, can the graph of $P(X)$-in the closed interval of real numbers whose end-points are $X(1)$ and $X(n)$ always be a subset of $T$? I have been unable to find proofs or counter-examples to answer either of these questions.

The triangle $T$ becomes of interest when dealing with data of the following sort. The $X$ values represent fractions or percentages of a population, each member of which has some yearly income. If the total yearly income of all the members of the population is $Q$, then the $Y$ value corresponding to each $X$ value represents that fraction of $Q$ which is the total yearly income of the poorest $X$-percent of the population.

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  • $\begingroup$ Please excuse this meaningless question. It is all that remains of a long question, all the rest of which got deleted. I have no idea what caused the deletion. $\endgroup$ – Garabed Gulbenkian Nov 8 '14 at 20:51
  • $\begingroup$ I think there were some initial spaces or tabs that caused it to be formatted so as to put everything on one line, most of which was not visible. It should all be visible now, though it would be better if you used TeX (enclose math in dollar signs). $\endgroup$ – Bjorn Poonen Nov 8 '14 at 20:55
  • $\begingroup$ Many thanks, Bjorn, for fixing the mess I made. I was trying to start a new line beginning with several spaces, so that it would look like the start of a new paragraph. Every time I try to do this, I get into trouble. $\endgroup$ – Garabed Gulbenkian Nov 8 '14 at 21:07

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