According to this site, if output price increases from $p*$ to $p'$ and factor prices remain constant, then a new production bundle chosen must yield at least the same amount of profits as the old bundle, as the old profit would yield at least as much profit in higher output price. The reasoning behind which is understandable.
However, how does it also apply for the case when output price decreases from $p*$ to e.g. $p"$? How does one be sure that the maximum profit at $p"$ would definitely be higher than the one at $p*$ (instead of lower)?
Also what does it mean by "reverse convexity" in "A reverse convexity will be obtained if we plot $\pi (p, w)$ against a particular factor price, $w_i$." in the aforementioned proof on the same site? I can not even find a proper definition of the term on internet.