2021/08/05 by Sudeshna Basu, Susmita Seal, Basu, Sudeshna +1
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2108.02908
openalex publication_date 2021/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we study three different versions of small diameter properties\nof the unit ball in a Banach space and its dual. The related concepts for all\nclosed bounded convex sets of a Banach space was initiated and developed in\n citeB3, citeBR , citeEW, citeGM was extensively studied in the\ncontext of dentability, huskability, Radon Nikodym Property and Krein Milman\nProperty in citeGGMS. We introduce the the Ball Huskable Property (BHP),\nnamely, the unit ball has relatively weakly open subsets of arbitrarily small\ndiameter. We compare this property to two related properties, BSCSP namely,\nthe unit ball has convex combination of slices of arbitrarily small diameter\nand BDP namely, the closed unit ball has slices of arbitrarily small\ndiameter. We show BDP implies BHP which in turn implies BSCSP and none of\nthe implications can be reversed. We prove similar results for the\nw^*-versions. We prove that all these properties are stable under lp sum\nfor 1\≤ p \≤ \∞, c0 sum and Lebesgue Bochner spaces. Finally, we\nexplore the stability of these with properties in the light of three space\nproperty. We show that BHP is a three space property provided X/Y is finite\ndimensional and same is true for BSCSP when X has BSCSP and X/Y is\nstrongly regular ( citeGGMS).\n