By Rolf Schneider

On the middle of this monograph is the Brunn-Minkowski idea, which are used to nice impact in learning such rules as quantity and floor zone and their generalizations. specifically, the notions of combined quantity and combined sector degree come up certainly and the basic inequalities which are happy through combined volumes are thought of right here intimately. the writer provides a entire advent to convex our bodies, together with complete proofs for a few deeper theorems. The publication offers tricks and tips to connections with different fields and an exhaustive reference record. This moment version has been significantly multiplied to mirror the speedy advancements of the earlier 20 years. It comprises new chapters on valuations on convex our bodies, on extensions just like the Lp Brunn-Minkowski concept, and on affine structures and inequalities. There also are many vitamins and updates to the unique chapters, and a considerable growth of bankruptcy notes and references

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Xn+1 are aﬃnely independent. Put λ1 + · · · + λn+1 =: λ and n+1 y := i=1 λi xi . 12 gives y ∈ int conv {x1 , . . , xn+1 } ⊂ int P. If k = n + 1, then x = y ∈ int P. Otherwise, put k z := λi xi . 9. 15 Let A ⊂ Rn be convex. Then (a) relint A = relint cl A, (b) cl A = cl relint A, (c) relbd A = relbd cl A = relbd relint A. Proof We may assume that dim A = n. Part (a): trivially, int A ⊂ int cl A. Let x ∈ int cl A. Choose y ∈ int A. 9 shows that x ∈ int A. Part (b): trivially, cl A ⊃ cl int A. Let x ∈ cl A.

2 Let D ⊂ Rn be convex and let f : D → R be a continuous function. Suppose that for each point x0 ∈ D there are an aﬃne function g on Rn and a neighbourhood U of x0 such that f (x0 ) = g(x0 ) and f ≥ g in U ∩ D. Then f is convex. Proof Since a function is convex if its restriction to any segment is convex, it suﬃces to consider the case where n = 1 and D is an interval in R. Suppose that f satisfies the assumption but is not convex. Then there are points x1 < z < x2 in D and an aﬃne function h on R such that f (x1 ) = h(x1 ), f (x2 ) = h(x2 ) and f (z) > h(z).

In fact, for suitable ρ > 0 the set B(x, ρ) ∩ A is compact and nonempty, hence the continuous function y → |x−y| attains a minimum on this set, say at y0 ; then |x−y0 | ≤ |x − y| for all y ∈ A. If also y1 ∈ A satisfies |x − y1 | ≤ |x − y| for all y ∈ A, then z := (y0 + y1 )/2 ∈ A and |x − z| < |x − y0 |, except if y0 = y1 . Thus, y0 =: p(A, x) is unique. In this way, a map p(A, ·) : Rn → A is defined; it is called the metric projection or nearest-point map of A. It will play an essential role in Chapter 4, when the volume of local parallel sets is investigated.