Schnittvolumina hochdimensionaler konvexer Körper

This thesis deals with volumes of hyperplane sections of convex and star-shaped bodies. In the first part we study the extremal volumes of central hyperplane sections of unit balls in the space l_p^n(l_2^m) for all p ∈ (0,2] ∪ {∞} and m, n ∈ IN. To this end we give multiple formulas for volumes and use them to show that the normal vector (1,0,...,0) generates the minimal volume in the case p = ∞. Furthermore we give an upper bound for the (n−1)-dimensional volumes for m ∈ IN≥3 and p = ∞. For n → ∞ this bound is attained for the normal vector (1/√n,...,1/√n). Another central result of this thesis is the statement that the normal vector (1/√n,...,1/√n) minimizes and that (1,0,...,0) maximizes the volume for p ∈ (0,2]. In the second part we consider the extremal volumes of slabs in the unit ball in the space l_∞^n(l_2^m). We give volume formulas for slabs to prove that the volume of the slab orthogonal to (1,0,...,0) with width t is minimal for all t ∈ [0,(m+2)/(m+3)] and m ∈ IN≤3. Moreover we discuss the case m ∈ IN≥4. Futhermore we prove the following fact: The volume of the slab orthogonal to (1/√2,1/√2,0,...,0) with width t is smaller than the volume of the slab orthogonal to (1/√n,...,1/√n) with width t for all arbitrary small t ∈ IR≥0, for all m ∈ IN≥3 and almost all n ∈ IN. For m = 1 we show an analogous result to that obtained for the estimation of the volumes of sections: The volume of a slab with very small width orthogonal to (1/√2,1/√2,0,...,0) is maximal compared to all other volumes of slabs with corresponding width and a large class of normal vectors. For m = 2 we obtain that for a slabs of width t and normal vector (1/√2,1/√2,0,...,0) the volume is smaller than the volume for a slab of width t and normal vector (1/√n,...,1/√n) for all t ∈ [1/n, 5/11]. In the third part we study volumes of non-central sections and slabs in the unit ball of the space l_∞^n(l_2^1) more detailed. We get further volume formulas and additionally find the global minimal and maximal values for the 2- and 3-dimensional l_1-ball given the distance to the origin respectively the width of the slab. In the last part of this thesis we consider sections through the centroid of the n-dimensional regular simplex. We state a volume formula and show that √(n+1)/(n-1)!*1/√(3e)*1/e is a lower bound for the volume of sections through the centroid.