T(X ⊕₁ Y) equals min{T(X),T(Y)} when one summand has the Daugavet property, but there exist spaces X = D ⊕_N D where T(X ⊕₁ X) < T(X).
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A counterexample for the Daugavet index of thickness in $\ell_1$-sums
T(X ⊕₁ Y) equals min{T(X),T(Y)} when one summand has the Daugavet property, but there exist spaces X = D ⊕_N D where T(X ⊕₁ X) < T(X).