Characterization of real inner product spaces by Hermite-Hadamard type orthogonalities

In this study, we provide some new characterizations of the real inner product spaces using the notion of Hermite-Hadamard (HH) type orthogonalities and by considering their relationships with Birkhoff-James orthogonality. In addition, we investigate the classes of linear mappings that preserve two special types of these orthogonalities. In particular, we show that every HH-I-orthogonality preserving linear mappings is necessarily a scalar multiple of a linear isometry. Finally, we present some other characterizations of real inner product spaces in terms of HH-P-and HH-I-orthogonality preserving mappings. (C) 2019 Elsevier Inc. All rights reserved.