# Is this min not less than a min

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

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Thanks in advance for any correct answer or any help that can lead to a correct answer. – userior Feb 24 '13 at 7:19
or is there any counterexample? Thanks. – userior Feb 24 '13 at 7:41
If it's not too much trouble, could you please explain where you ran into this problem and whether or not there are partial results that you have already found? – Yemon Choi Feb 24 '13 at 9:44
1. perimeter/area = 2/inradius; so there is an equivalent formulation with "max min inradius($v_iv_jv_k$)". 2. Both quantities are constants. Wouldn't you rather know the value of those constants? – Günter Rote Feb 24 '13 at 19:18
@partial results: A natural conjecture is that the second quantity is minimized when the 5 points form a regular pentagon. Have you checked this? What is the value for a regular pentagon? Is the first quantity perhaps minimized for a square plus center? What is the value for this configuration? What is the best value that you know for each of these quantities? – Günter Rote Feb 24 '13 at 19:25

$$\max_{v_{0},v_{1},v_{2},v_{3},v_{4}\in\partial\mathbf{D}} \min_{0\le i,j,k\le4} \mbox{inradius}(v_i,v_j,v_k).$$
If we don't have a regular pentagon, then there are 3 points who are contained in an arc whose length is strictly less than $4\pi/5$, suppose they are $v_1,v_2,v_3$ in this order. The $\mbox{inradius}(v_1,v_2,v_3)$ is maximized if $v_2$ is halfway between $v_1$ and $v_3$. So this quantity will be less than the inradius of three consecutive vertices of the pentagon, which is less than the inradius of three non-consecutive vertices of the pentagon.