Prove that $S(t)$ is bounded

Prove that $S(t)$ is convex. That is, if $A,B$ are in $S(t)$ then $AB$ is in $S(t)$.

Identify all points $X$ on $S(t)$ such that $PX+QX$ is minimum.

Prove that the area of $S(t)$ is a convex function of $t$.

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## Friday, September 19, 2014

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Inverse distances to points

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Math is evil...

Let $P$ and $Q$ be two distinct points on a plane. For any given point $X \neq P,Q$, define two functions $f(X)$ and $g(X)$ as follows:
$$f(X) = \frac{1}{PX} + \frac{2}{QX}$$
$$g(X) = \frac{2}{PX} + \frac{1}{QX}$$
Now for any positive number $t$, let $S(t)$ be the set of all points $Y$ such that $f(Y) \leq tg(Y)$.

Prove that $S(t)$ is bounded

Prove that $S(t)$ is convex. That is, if $A,B$ are in $S(t)$ then $AB$ is in $S(t)$.

Identify all points $X$ on $S(t)$ such that $PX+QX$ is minimum.

Prove that the area of $S(t)$ is a convex function of $t$.

Labels:
analytic,
convex,
convex hull,
Geometry,
Inequality,
projective geometry

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