By Patrick Suppes (Auth.)
Foundations of dimension bargains the main coherently equipped remedy of the subjects and matters critical to size. a lot of the study concerned has been scattered over numerous many years and a mess of journals--available often basically to specialties. With the book of Volumes and 3 of this crucial paintings, Foundations of dimension is the main accomplished presentation within the region of size
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Extra info for Foundations of Measurement. Geometrical, Threshold, and Probabilistic Representations
Example text
The most familiar examples of norms are the i f by the equation (for 1 < ρ < oo) p norms, given in Re" l/P /= 1 or, in certain vector spaces of integrable functions, by the analogous equation = \ \f(u)Uu For J? norms with 1 < ρ < oo, strong convexity holds; this result is the well-known Minkowski inequality. ) The space (Re", + , ·>, with its ordered p affine structure and an S£ norm, is the standard example of Η-dimensional Minkowski geometry. For ρ — 2, we have the Euclidean norm, whose 2 2 2 special feature is that the function (l/2)[||x + >>|| - ||JC|| - ||>>|| ] is bilinear and, hence, can be taken as a scalar product; the analogous construction fails to be bilinear for ρ Φ 2.
Consider ordinary three-dimensional space, and choose an arbitrary point 0 as an origin from which to project and an arbitrary plane Ρ that does not contain 0 as our "projective" plane. 2. 25 VECTOR REPRESENTATIONS Ox F I G U R E 1. 0 is the origin in 3-space. Ox and 0 ^ are projective points corresponding to affine points x, y in the plane P; OL is the projective line determined by Ox and Oy corresponding to the affine line L in P. From the standpoint of P, is the projective line at infinity, the plane through 0, parallel to P.
We consider two kinds of geometry. Straight spaces are those in which the metric is defined in an open subset X of an Η-dimensional affine geometry, and the portion of any affine line that intersects X is isometric to a Euclidean line. Spaces of elliptic type are those in which the metric is defined on the entire w-dimensional projective geometry, and each projective line is isometric to a Euclidean circle. ) We give just the definitions of these two types, together with a few general remarks, and then provide examples of each: the Hilbert geometries, one of which can be defined in the interior of any closed convex surface in affine space, and which include the hyperbolic geometries, defined in the interiors of ellipsoids; and the classical elliptic geometry, defined on the entire projective space.
