Let the sequence of convex $n$-gons $\{P_k\}$. Is there a subsequence $\{P_{k_i} \}$ of that sequence such that $P_{k_i} \to P$ as $k_n \to \infty $, where $P$ is a $m$-gon for $m \leq n$ ? I think the Blaschke Selection Theorem is an option for the resolution of that problem, but I am not quite sure how to resolve with that result. Could anyone be able to help me at this point? Particularly, why $P$ must be an $m$-gon in such a way that $m \leq n$?
Blaschke Selection Theorem : For a sequence $\{K_n\}$ of convex sets contained in a bounded set, there exists a subsequence $\{K_{n_m}\}$ and a convex set $K$ such that $K_{n_m}$ converges to $K$.
Clarification : The limit is endowed with the usual Hausdorff metric.
In general it is not true since you can take $P_k = kP$ where $P\neq \{0\}$ is any $n-$ gon. But if you assume that there exist a ball $B$ such that $P_k \subset B$ for infinitely many $k.$ Then it is true. To see this let $A_k =\{p_{k1} , p_{k2} , ... p_{kn}\} $ be the set of vertices of the $n-$ gon $P_k .$ Then there exists a sequence of natural numbers $s_j $ such that $p_{s_j l} \to p_l $ for all $1\leq l\leq n$ as $j\to \infty $, where $p_l \in \mathbb{R}^d .$ From the inequlity $$d_H (\mbox{conv} (A) , \mbox{conv}(B) ) \leqslant d_H (A,B)$$ we obtain that $P_{s_j } \to \mbox{conv}\{ p_1 , p_2 , ..., p_n\} $ as $i\to \infty$ with respect to the Hausdorff metric $d_H .$