Existence of diametrically complete sets with empty interior in reflexive and separable Banach spaces

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

“…Thus our paper is a complement to the papers [7], [24] and [39]. More precisely, in [7] it is proved that such a norm exists for each infinite-dimensional, separable and reflexive Banach space.…”

“…Thus our paper is a complement to the papers [7], [24] and [39]. More precisely, in [7] it is proved that such a norm exists for each infinite-dimensional, separable and reflexive Banach space. In [24] we extend this result and prove that for each infinite-dimensional reflexive Banach space (X, • X ) with the nonstrict Opial and the Kadec-Klee properties, there exists an equivalent norm such that X equipped with this norm is LUR and contains a diametrically complete set with empty interior.…”

Renormings of Nonseparable Reflexive Banach Spaces and Diametrically Complete Sets with Empty Interior

“…Now, we are ready to prove the following theorem regarding the nonstrict Opial property. The proof is a modification of the proof of Theorem 5.4 in [5] (see also [3,4]) and therefore we omit it. We also note that the main idea of the proof is due to Maluta [17].…”

“…Remark 6.3. It is known that every separable Banach space (X, • X ) can be equivalently renormed in such a way that this space with the new norm has both the nonstrict Opial and the Kadec-Klee properties (see [2,5]). An analogous result for nonseparable and reflexive Banach spaces is not known.…”

“…It has recently been proved in [5] that after a suitable renorming, each infinite-dimensional, reflexive and separable Banach space is locally uniformly rotund and contains a diametrically complete set with empty interior. In the present paper we extend this result to each nonseparable and reflexive Banach space which has the nonstrict Opial and Kadec-Klee properties.…”