Eisenhart L. P.'s Affine Geometries of Paths Possessing an Invariant Integral PDF

By Eisenhart L. P.

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Yp }) ∪ cone({v1 , . . , vq }))∗ = P (Y , 1; V , 0), with P (Y , 1; V , 0) = {x ∈ Rd | Y x ≤ 1, V x ≤ 0}. Conversely, given any p × d matrix, Y , and any q × d matrix, V , we have P (Y, 1; V, 0)∗ = conv({y1 , . . , yp } ∪ {0}) ∪ cone({v1 , . . , vq }), where yi ∈ Rn is the ith row of Y and vj ∈ Rn is the j th row of V or, equivalently, P (Y, 1; V, 0)∗ = {x ∈ Rd | x = Y u + V t, u ∈ Rp , t ∈ Rq , u, t ≥ 0, Iu = 1}, where I is the row vector of length p whose coordinates are all equal to 1.

SEPARATION AND SUPPORTING HYPERPLANES (where inf is the notation for least upper bound). Now, if X is an affine space of dimension d, it can be given a metric structure by giving the corresponding vector space a metric structure, for instance, the metric induced by a Euclidean structure. , so that d(a, S) = d(a, s). The proof uses the fact that the distance function is continuous and that a continuous function attains its minimum on a compact set, and is left as an exercise. 10 Given an affine space, X, let A and B be two nonempty disjoint closed and convex subsets, with A compact.

Therefore, H separates A and {a}. Remark: The assumption that A is closed is convenient but unnecessary. 17 shows that the proposition holds for every boundary point, a ∈ ∂A (assuming ∂A = ∅). 17 is false when the dimension of X is infinite and when A= ∅. The proposition below gives a sufficient condition for a closed subset to be convex. 18 Let A be a closed subset with nonempty interior. If there is a supporting hyperplane for every point a ∈ ∂A, then A is convex. Proof . 4). The condition that A has nonempty interior is crucial!

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Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P.


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