functional analysis

Hereditary Diameter Rigidity in Real -Spaces

arXiv:2607.12770

summary

The paper shows that lower bounds on the hereditary diameter of sets in real L₁ spaces remain stable when taking convex combinations, giving explicit constants, and uses this to prove equivalences between diameter‑two and strong diameter‑two properties (and related regularity properties) on bounded convex subsets of the unit ball.

Abstract

We prove that hereditary diameter lower bounds are stable under convex combinations in real -spaces over arbitrary measure spaces. More precisely, if every weakly open piece of each set has diameter at least , then every finite or countable convex combination has the corresponding hereditary lower bound . 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.

16 pages, 0 figures

Topics & keywords

#real l1 spaces#hereditary diameter#convex combinations#diameter two property#weak topologyhereditary diameter lower boundconvex combination stabilityreal L1-spacediameter two propertystrong diameter two propertyweak topology
Hereditary Diameter Rigidity in Real $L_1$-Spaces · wovepaper