By Michael K. Murray, John W. Rice (auth.)

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The second comment is that in Barndorff-Nielsen (1978) notions of convexity for subsets of Euclidean space, play an important role. All these notions can also be applied to a finite-dimensional affine space. Given any two points p and q in an affine space there is a THE GEOMETRY OF EXPONENTIAL FAMILIES 24 = vector v such that q P + v. The line segment joining p and q is then defined to be the set [p, q] = {p + tv 1 t E [0, I]} and a set is said to be convex if whenever it contains two points it contains the line segment joining them.

N 0,)'(0) are to be, which by definition are the values d¢l(v), ... ,d¢n(v). In this case one can calculate dl( v) using the chain rule which, from the definition of differential, assumes the form dl(v) 01 oj = O¢l (p)d¢I(V) + ... + o¢n (p)d¢n(v) The chain rule thus becomes an identity between functions of tangent vector, and if we leave out the argument of the functions, as is usual when expressing an identity between functions, we obtain the formula for the so-called total differential 01 1 + ...

These are the variations for which all the coordinate functions are held constant except one, which changes at a unit rate. Specifically, if ¢} , ... 2 THE BASIC APPARATUS 31 set P and pEP then we let Ii (h) be the element of P such that We call 'Yi the ith co-ordinate variation through p. (See Figure. 2). These are exactly the variations used to define the partial derivatives of f with respect to