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Volume estimates for sections of certain convex bodies
Author(s) -
Brzezinski Patryk
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200119
Subject(s) - mathematics , hyperplane , simplex , convex body , mixed volume , upper and lower bounds , volume (thermodynamics) , regular polygon , combinatorics , euclidean space , centroid , geometry , mathematical analysis , unit sphere , convex hull , physics , quantum mechanics
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 → ∞ .