Some remarkable lines of triangles in real normed spaces and characterizations of inner product structures

Characterizations of inner product spaces by geometrical properties of the heights in a triangle

“…In [1] we defined ªthe height vectorsº hxY y y kyk 2 À & H xY y kx À yk 2 x À y and HxY y y kyk 2 À & H yY x kx À yk 2 x À y 1…”

“…The norm of hxY y is a generalization of the usual height in an inner product space (i.p.s.). Using this heights definition we have obtained (jointly with C. Alsina), different characterizations of inner product spaces (see [1], [2] and [3]). …”

On parallelogram areas in normed linear spaces

“…h is the height vector (see [1]) given by h(x, y) = y + y 2 − ρ ′ + (x, y) x − y 2 (x − y) with x = y because x, y are independent and s is the semiperimeter of the triangle with vertices 0, x, y:…”

Incenters in real normed spaces

“…In [1] we generalized the concept of the bisectrix of an angle in euclidean spaces by considering in E the vectors b ± (x, y) = y x y + x k ± (y, x) +…”

“…Using the above definitions of bisectrices, jointly with C. Alsina and P. Guijarro we have obtained different characterizations of inner product spaces (see [1,3]). …”