Abstract: The minimal widths of three bounded subsets of the unit sphere associated to a unit vector in a normed linear space are studied, and three related geometric constants are introduced. New characterizations of inner product spaces are also presented. From the perspective of minimal width, strong $\varepsilon$-symmetry of Birkhoff orthogonality is introduced, and its relation to $\varepsilon$-symmetry of Birkhoff orthogonality is shown. Unlike most of the existing parameters of the underlying space, these new constants are full dimensional in nature.

Key words: Birkhoff orthogonality, isosceles orthogonality, minimal width, Singer orthogonality, strong $\varepsilon$-symmetry of Birkhoff orthogonality, width function