Following the volume difference function, we first introduce the notion of the affine surface area quotient function. We establish Brunn–Minkowski type inequalities for the affine surface area quotient function, which in special cases yield some well-known results. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 1022 – 1029 ¨ ITAK ˙ c TUB ⃝ doi: On quotients of ith affine surface areas Chang-Jian ZHAO∗ Department of Mathematics, China Jiliang University, Hangzhou, . China Received: • Accepted: • Published Online: • Printed: Abstract: Following the volume difference function, we first introduce the notion of the affine surface area quotient function. We establish Brunn–Minkowski type inequalities for the affine surface area quotient function, which in special cases yield some well-known results. Key words: Volume difference function, affine surface area quotient function, Blaschke sum, Brunn–Minkowski inequality 1. Introduction and statement of results The well-known classical Brunn–Minkowski inequality can be stated as follows: If K and L are convex bodies in Rn , then (see, ., [16]) V (K + L)1/n ≥ V (K)1/n + V (L)1/n , () with equality if and only if K and L are homothetic. Here, + is the usual Minkowski sum. Let K and L be star bodies in Rn , then the dual Brunn–Minkowski inequality states that (see [8]) ˜ 1/n ≤ V (K)1/n + V (L)1/n , V (K +L) () ˜ is the radial Minkowski sum. with equality if and only if K and L are dilates. Here, + A vector addition was defined on Rn which we call radial Minkowski addition, as follows. If x, y ∈ Rn , ˜ is defined to be the usual vector sum of x, y provided x, y both lie in a 1 -dimensional subspace of then x+y n R and as the zero vector otherwise. If K, L are star bodies and λ, µ ∈ R, then the radial Minkowski linear ˜ ˜ ˜ ˜ is called the combination, λK +µL , is defined by λK +µL = {λx+µy : x ∈ K, y ∈ L}. The expression K +L radial Minkowski sum of the star bodies K and L (see [5]). In 2004, Leng [6] defined the volume difference function of compact domains D and K , where D ⊆ K , by DV (K, D) = V (K) − V (D). The following Brunn–Minkowski .
