Hereditary Diameter Rigidity in Real $L_1$-Spaces

We prove that hereditary diameter lower bounds are stable under convex combinations in real $L_1$-spaces over arbitrary measure spaces.

More precisely, if every weakly open piece of each set has diameter at least $\delta>0$, then every finite or countable convex combination has the corresponding hereditary lower bound $\delta/4$.

In the diameter-two case, the bound is preserved exactly.

Consequently, for every topology containing the relative weak topology, the diameter two and strong diameter two properties are equivalent on bounded convex subsets of the unit ball, as are the convex point of continuity property and strong regularity.