Combinatorial Analysis by R. Bellman, M. Hall (ed.)

By R. Bellman, M. Hall (ed.)

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Characterize those convex bodies K for which ddx + y) = ddx) implies that x and y are multiples. + dK(y) 6. 1I with respect to the unit sphere S := {x (x, x) = I}. 1I with fj. 1I , then, ° ° rr(u) = Hu := {x I (x, u) = I}. If the affine subspaces U and V which generate W are not parallel and if W does not contain 0, then, rr(W) = rr(U) n rr(V). Note that rr 0 rr is the identity. n by adding a "hyperplane at infinity", Hoo. Then, rr(O) = Hoo. That will be needed, for example, in Lemma 3. 1 Definition.

We show that d K is well-defined (part (b) of the following lemma). 11 Lemma. Let K be an n-dimensional convex body in lRn. (a) If a line g intersects aK in three different points, then, g is contained in a supporting hyperplane of K, so, in particular, g n int K = 0. (b) Any ray emanating from a point in int K intersects aK in one and only one point. PROOF. (a) Let A, B, C E g n aK, and let B lie between A and C. IfHdidnotcontain Exercises 23 both A and C, it would separate these points properly, which contradicts the definition of a supporting hyperplane.

We write L' V* U, ---+ U*. ~ bv of V and the dual basis br, ... , b~ of V*, so that b* (b) i j £ = Uij:= ° {I for i for i = j, "# j. For every subset M of V, there is a vector space Ml. := {x* E V* I x*(M) = OJ. For mEV, we write ml. instead of {m}l.. Now, we will introduce the notion of a short, exact sequence. Assume we are given a sequence (1) of linear maps and vector spaces. Then, we call (1) a short, exact sequence if LI is surjective, L2 is injective, and ker LI = im L 2. Then, we may interpret W as a linear subspace of V and L2 as the injection W '-+ V.

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