Generalized Daugavet equations, affine operators and unique best approximation
We introduce and investigate the notion of generalized Daugavet equation $\| A_1+\cdots +A_n\| =\| A_1\| +\cdots +\| A_n\| $ for affine operators $A_1,\ldots ,A_n$ on a reflexive Banach space into another Banach space. This extends the well-known Daugavet equation $\| T+I\| =\| T\| +1$, where $I$ denotes the identity operator. A new characterization of the Daugavet equation in terms of extreme points is given. We also present a result concerning uniqueness of best approximation.