For dual quermassintegrals, the Brunn-Minkowski inequality fails for q>n (and q>n+2 for origin-symmetric bodies), holds at q=n+2 for origin-symmetric bodies, and holds for 0<q≤n+1 for unconditional bodies.
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On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$
For dual quermassintegrals, the Brunn-Minkowski inequality fails for q>n (and q>n+2 for origin-symmetric bodies), holds at q=n+2 for origin-symmetric bodies, and holds for 0<q≤n+1 for unconditional bodies.