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

Abstract

We show that each infinite-dimensional reflexive Banach space $$(X,\left\| \cdot \right\| _X)$$ ( X , · X ) has an equivalent norm $$\left\| \cdot \right\| _{X,0}$$ · X , 0 such that $$(X,\left\| \cdot \right\| _{X,0})$$ ( X , · X , 0 ) is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space $$C([0,1],{\mathbb {R}})$$ C ( [ 0 , 1 ] , R ) contains a constant width set with empty interior.