The best thing in the festival could have been enjoyed in the cool quiet of one's study : Painter's book . ( Mahler, one feels constrained to add, means painter in German . ) Divided into four sections, it offers essays imbedding Mahler in his historical context, more formal musical analyses and two sections of examples of critical response to Mahler in Europe and the United States.
42.
Given a compact manifold " X " imbedded in, say, Euclidean space " E ", we could define the filling radius " relative " to the imbedding, by minimizing the size of the neighborhood U _ \ varepsilon X \ subset E in which " X " could be homotoped to something smaller dimensional, e . g ., to a lower-dimensional polyhedron.
43.
The proof of this relies upon an improved regularity theorem that says that if \ partial \ Omega is C ^ k and f \ in H ^ { k-2 } ( \ Omega ), k \ geq 2, then u \ in H ^ k ( \ Omega ), together with a Sobolev imbedding theorem saying that functions in H ^ k ( \ Omega ) are also in C ^ m ( \ bar \ Omega ) whenever 0 \ leq m.
44.
Given a doubly periodic metric on \ mathbb R ^ 2 ( e . g . an imbedding in \ mathbb R ^ 3 which is invariant by a \ mathbb Z ^ 2 isometric action ), there is a nonzero element g \ in \ mathbb Z ^ 2 and a point p \ in \ mathbb R ^ 2 such that \ operatorname { dist } ( p, g . p ) ^ 2 \ leq \ frac { 2 } { \ sqrt { 3 } } \ operatorname { area } ( F ), where F is a fundamental domain for the action, while \ operatorname { dist } is the Riemannian distance, namely least length of a path joining p and g . p.
