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I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says

It's not obvious how to interpret the exponential function in (R',min,+) but it turns out that the natural choice is to consider the ordinary linear functions (in the conventional sense) to be the correct analogue.

So why are ordinary linear functions a natural analogue for the exponential function in a tropical semiring?

(Note: an intuitive answer is fine. I don't know any tropical geometry and only basic algebraic geometry, so not necessarily looking for a very technical response, though that would be awesome as well.

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