Stability Theorems in Geometry and Analysis

1. Preliminaries, Notation, and Terminology n n 1.1. Sets and functions in lR. - Throughout the book, lR. stands for the n-dimensional arithmetic space of points x = (X}, X2, '", xn)j Ixl is the length of n n a vector x E lR. and (x, y) is the scalar product of vectors x and y in lR., i.e., for x = (Xl, X2, -.-, xn) and y = (y}, Y2, --., Yn), Ixl = Jx + x + ... + x, (x, y) = XIYl + X2Y2 + ... + XnYn. n Given arbitrary points a and b in lR., we denote by [a, b] the segment that joins n them, i.e. the collection of points x E lR. of the form x = >.a + I'b, where>. + I' = 1 and >. 0, I' O. n We denote by ei, i = 1,2, ..., n, the vector in lR. whose ith coordinate is equal to 1 and the others vanish. The vectors el, e2, ..., en form a basis for the space n lR., which is called canonical. If P( x) is some proposition in a variable x and A is a set, then {x E A I P(x)} denotes the collection of all the elements of A for which the proposition P( x) is true.