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I have often wondered whether there has ever come a point in your research,

when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\subset\mathbb{R}^n$,

you checked on a computer that the function is likely to be non-negative on that entire set,

but you had enormous difficulty proving it?

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Sorry if this question appears too "soft". I am not at research-level yet, having just completed my undergraduate degree.

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  • $\begingroup$ There are problems like "what is the maximal area of an $n$-gon with diameter 1" (open already for $n=10$, as I am concerned), which, once you get an example, are reduced to proving an inequality for a continuous function on a finite dimensional compact set. If you allow $n$ to be not fixed, then the variety of problems of your kind enlarges dramatically. $\endgroup$ Apr 29 at 6:17
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    $\begingroup$ Agree with @Fedor. Part of the difficulty is, what counts as “explicit” and “explicitly defined”? Both the function and the domain over which you are trying to minimize it can be “explicit” and yet hugely difficult to understand. See e.g. Sendov’s conjecture (also open for $n$ as little as 12 or 13 IIRC) for another example of this phenomenon. $\endgroup$
    – Dan Romik
    Apr 29 at 7:09
  • $\begingroup$ Thanks @DanRomik after a bit of Googling I've now also found these other conjectured inequalities concerning matrices mdpi.com/2073-8994/13/10/1782/pdf $\endgroup$ Apr 29 at 10:38

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