Powered distances

Let $p$ and $q$ be two infinitely long non-parallel lines on a plane. For any point $X$, let $d_p(X), d_q(X)$ denote distances from $X$ to $p$ and $q$ respectively. For any positive number $r$, let $S(r)$ be the set of all points $Y$ such that $d_p(Y)^{2015} + d_q(Y)^{2015} \leq r$.

Prove that $S(r)$ is bounded.

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

Identify all points in $S$ such that $d_p(X)^{2014} + d_q(X)^{2014}$ is maximum.

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

3 Sequence Inequality

If $a_i, b_i, c_i$ are sequences of positive numbers for $i = 1,2,\cdots,n$, prove the following inequality:

$\sum (a_i+b_i+c_i) \sum \frac{a_ib_i + b_ic_i+ c_ia_i}{a_i+b_i+c_i} \sum \frac{a_ib_ic_i}{a_ib_i + b_ic_i+ c_ia_i} \leq \sum a_i \sum b_i \sum c_i$

where all summations are taken from $i=1$ to $i=n$. When does equality happen?