Understanding Legendre-fenchel Transform, looking for an easy example and intuition

Question

Looking for help in understanding this transform. I have no background in real analysis but need this stuff for my research.

I hope someone can give me some light on the intuition behind this transform and better if you can provide some example I can work with.

Thanks in advance.

I already found some information related to it here

But, still not able to fully understand.

Thanks.

Answer 1

Will answer my own question since I found a very good reading material which helped me to fully understand the topic.

If interested, take a look here.