Coincidence Functions and Their Integrals
Herbert Fédérer · Transactions of the American Mathematical Society · 1946
Introduction.Two functions/and g are said to coincide at (x, y) if and only if f(x) =g(y).If/ parametrizes a ¿-dimensional surface in Euclidean «-space, and g an (» -¿)-dimensional surface, then the number of coincidences of / and g is the number of intersections of these surfaces.Now keep the first surface fixed and move the second rigidly; or, otherwise said, superimpose upon g an arbitrary isometric transformation, S, of «-space.Count the number of coincidences of/with the superposition (S:g), that is, the number of intersections of the fixed and the movable surface.Integrate this count over the group of all isometric transformations of «-space with respect to its Haar measure, properly and explicitly normalized.If/and g are sufficiently regular, say Lipschitzian, the value of the integral is the product of the areas of the two surfaces times a number, ß(n, k), which depends only on the dimensions « and k.Poincaré proved this for two curves in the plane, the special case « = 2, k=n -k = l (see [P ])(*).For a curve and a line segment it had previously been obtained by Crofton.The "integral geometry" of Blaschke and his students is related to the problem (see [B]).In fact Santaló obtained our result for a curve and an ordinary surface in 3-space, the special case » = 3, k = l, n-k = 2 (see [S]).