New PDF release: A course on convex geometry

By Weil W.

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This is equivalent to K ⊂ B(c ), for all K ∈ M, for some constant c > 0. Here, we can replace the ball B(c ) by any compact set, in particular by a cube W ⊂ Rn . The subset M is relative compact, if every sequence K1 , K2 , . . , with Kk ∈ M, has a converging subsequence. Therefore, the mentioned topological property is a consequence of the following theorem. 4 (Blaschke’s Selection Theorem). Let M ⊂ Kn be an infinite collection of convex bodies, all lying in a cube W . Then, there exists a sequence K1 , K2 , .

With (j) (j) Wj (K1 ) = Wj (K2 ) = · · · , for all j ∈ N (j ≥ 2). Since √ min d(x, y) ≤ (j) y∈Kl n 2j , (j) for all x ∈ Kk , we have √ (j) (j) d(Kk , Kl ) ≤ n 2j , for all k, l ∈ N, and all j. By the subsequence property we deduce √ n (j) (i) d(Kk , Kl ) ≤ i , for all k, l ∈ N, and all j ≥ i. 2 (k) In particular, if we choose the ’diagonal sequence’ Kk := Kk , k = 1, 2, . . , then √ n d(Kk , Kl ) ≤ l , for all k ≥ l. 2 Hence (Kk )k∈N is a Cauchy sequence in M. Let ˜ k := cl conv (Kk ∪ Kk+1 ∪ · · · ) K and ∞ ˜ k.

B) For each u ∈ S n−1 , there is a supporting hyperplane E(u) of K (in direction u). Let A(u) be the open half-space of E(u) which fulfills A(u) ∩ K = ∅ (A(u) has the form { ·, u > hK (u)}). Then, the family {A(u) : u ∈ S n−1 } is an open covering of the compact set bd (K + B(ε)), since every y ∈ bd (K + B(ε)) fulfills y ∈ / K and is therefore separated from K by a supporting hyperplane E = E(u) of K. Therefore there exist u1 , . . , um ∈ S n−1 with m bd (K + B(ε)) ⊂ A(ui ). i=1 Let H(ui ) := Rn \ A(ui ) be the corresponding supporting half-space and m P := H(ui ), i=1 then K ⊂ P.

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A course on convex geometry by Weil W.


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