Let the sequence of convex bounded $n$-gons $\{P_k\}$. Is there a subsequence $\{P_{k_n}\}$ of that sequence such that $P_{k_n} \to P$ 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?
Clarification : The limit is endowed with the usual Hausdorff metric.
On
If the sequence is contained in a compact ball of $\Bbb R^t$, then the answer is yes: the sequence of vertices $(V_k^1,\cdots,V_k^n)$ is a bounded sequence in $\Bbb R^{n\times t}$. Therefore, it has a convergent subsequence $(V_{k_s}^1,\cdots,V_{k_s}^n)\to (H^1,\cdots, H^n)$. Now, show as an (a bit tedious, if I recal correctly) exercise that, if the vetices of a sequence of convex $n$-gons converge to $(H^1,\cdots, H^n)$ - not necessarily dinstinct -, then the sequence of $n$-gons converges in the Hausdorff metric to the convex hull of $H^1,\cdots, H^n$.
If your $P_k$ are all contained in a compact set $K$, and you allow degenerate cases such as single points or line segments to count as $m$-gons, then the answer is yes. By the Blaschke Selection Theorem, the space of nonempty closed subsets of $K$ with the Hausdorff metric is compact.