Using notions of Minkowski geometry (i.e., of the geometry of finite dimensional Banach spaces), the author derives the following characterization of planar centrally symmetric convex bodies: A convex body \(K\) in a Minkowski plane \(\mathcal M^2(B)\) is centrally symmetric iff every chord of \(K\) that bisects the Minkowskian perimeter of \(K\) is necessarily an affine diameter of \(K\). This theorem can be applied for deriving characterizations of centrally symmetric members within classes of special convex bodies (e.g., characterizations of unit Minkowskian balls within the class of planar bodies of constant Minkowskian width). The author presents some characterizations of equiframed curves and related characteristic properties of bodies of constant width and constant tangential width in Minkowski planes whose unit Minkowskian circle is an equiframed curve. In particular, the author shows that straightforward extensions of some properties of bodies of constant Euclidean width are also valid for bodies of constant Minkowskian width if the underlying Minkowskian circle is an equiframed curve. All obtained characterizations are restricted to the case of the plane and involve certain measures of boundary arcs that join antipodal points of a planar convex body.