AI 中文总结
研究实 \(L_1\) 空间中遗传直径下界在凸组合下的稳定性，证明相关性质，得出在特定拓扑下单位球有界凸子集上一些性质等价的结论。
AI 中文摘要
我们证明了在任意测度空间上的实 \(L_1\) 空间中，遗传直径下界在凸组合下是稳定的。具体而言，若每个集合的每个弱开子集直径至少为 \(\delta>0\)，则每个有限或可数凸组合有相应遗传下界 \(\delta/4\)。在直径为二的情况下，该界被精确保持。因此，对于包含相对弱拓扑的每个拓扑，单位球有界凸子集上直径二和强直径二性质等价，凸连续点性质和强正则性也等价。
英文摘要
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 $δ>0$, then every finite or countable convex combination has the corresponding hereditary lower bound $δ/4$. 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.
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