A linear programming problem, any textbook name or hint will be appreciated

Question

Let $a_1, a_2,a_3,a_4,d \in \mathbb{R}$ that $$a_3 < a_1 + d < a_4$$, $$a_1 < a_4 - d < a_2$$, $$a_1 < a_2 < a_3 < a_4$$

I tried to manipulate these inequalities to get some result. But I didn't find anything useful. How to find the minimum and maximum value of $d$?

The first inequality tells you $ a_3 - a_1 < d < a_4 - a_3$. What does the second inequality tell you about $d$? Hence, what is the min and max? — Calvin Lin, Feb 20 '23 at 22:23

yuanming luo · Accepted Answer · 2023-02-20T22:36:31.970

1

The first inequality implies $a_3 - a_1 < d < a_4 - a_1$, the second inequalities implies $a_4 - a_2 < d < a_4 - a_1$. Hence $d > \max\{a_3-a_1, a_4-a_2\}$. So the infimum value is $\max\{a_3-a_1, a_4-a_2\}$ and supremum value is $a_4 - a_1$.

edited Feb 20 '23 at 22:36

answered Feb 20 '23 at 22:31

yuanming luo

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A linear programming problem, any textbook name or hint will be appreciated

1 Answers1