Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m ‐codimensional sections of powers of m ‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n ‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m ‐codimensional sections x ∈ B ∞ , 2 : ∑ j = 1 n a j x j = 0 “perpendicular” to a ∈ S n − 1 ⊆ R nof unit balls B ∞ , 2in the spacel ∞ n ( l 2 m )for all m , n ∈ N . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the m ( n − 1 ) ‐dimensional volumes for natural numbers m ≥ 3 . This bound is asymptotically attained for the normal vector1 n , ⋯ , 1 nas n → ∞ .