I do not agree with the statement that volswap is short convexity or that varswap is long convexity etc. It depends what your 'base' instrument is: if it is the varswap then the volswap has convexity, if it is the volswap then the varswap has convexity. I.e. convexity is a relative measure. So to trade convexity (or "vol of vol" as the street calls it) you could trade long varswap short volswap (this would be a long vol of vol trade).
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