sobolev spaces – What is the optimal constant in an inequality?

My questions are that

  1. What is & # 39;optimal& # 39; constant in an inequality? and
  2. What is the difference between & # 39;sharp& # 39; and & # 39;optimal& # 39; constant?

In a paper theorem, it is given that

Let $ Omega subset mathbb R ^ n $ to be a convex domain of diameter $ d $ and $ u in W ^ {1,1} ( Omega) $ such as $ int limits _ { Omega} u = 0. $ then {{} {1} {2} {Green} {Green} { {1} {1} {1} {1}
In addition, the constant $ frac {1} {2} $ is optimal.

Any help is appreciated. Thank you.