Let $\mathbf{D}$ be the unit disk, is 
$$\min_{\begin{array}{c}
v_{1},v_{2},v_{3},v_{4}\in\mathbf{D}\\
v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right)
\end{array}}\max_{0\le i,j,k\le4}\frac{\mbox{perimeter}\left(\triangle v_{i}v_{j}v_{k}\right)}{\mbox{area}\left(\triangle v_{i}v_{j}v_{k}\right)}
$$no less than 
$$\min_{v_{0},v_{1},v_{2},v_{3},v_{4}\in\partial\mathbf{D}}\max_{0\le i,j,k\le4}\frac{\mbox{perimeter}\left(\triangle v_{i}v_{j}v_{k}\right)}{\mbox{area}\left(\triangle v_{i}v_{j}v_{k}\right)}?
$$