Vector-valued Maclaurin inequalities

Abstract

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds.

Isoperimetric problems for zonotopes

“…Similarly, Filliman [19] investigated the problem of finding the zonotopes of maximal volume with a fixed value of the squared lengths of its generating vectors. These results, Corollary 1, and the results of Brazitikos and McIntyre in [15] is the motivation behind our next definition.…”

“…The authors of [15] proved this conjecture for 𝑝 = 0 and 𝑝 = ∞, for 𝑝 = 2 and 𝑛 = 𝑑, and for 𝑝 = 1, 𝑛 = 𝑑 and 𝑘 = 2, 3, 𝑑. They also pointed out (see also Corollary 1) that if 𝑝 = 1, the numerators in (7) correspond to the 𝑘th and (𝑘 − 1)st intrinsic volumes of the zonotope…”

“…Our first goal is to prove a generalization of the decomposition theorem in [45], which yields an elementary proof of the formula in [15]. To state this result, we need some preparation.…”

“…In the paper, we also point out applications of our results to other problems in mathematics. We focus on two problems, one of which is the so‐called $$\ell _p$$ ‐polarization problem discussed, for example, in [3] (see also [39]), and the other one is a generalization of the well‐known Maclaurin inequality [10] by Brazitikos and McIntyre [15], comparing the values of elementary symmetric functions evaluated at positive real numbers. We introduce these problems and their relations to zonotopes in detail in Section 2, and describe our related results at their appropriate places in Sections 3 and 4.…”

“…Furthermore, for any 𝐼 ∈  𝑍 𝑘 with 𝑘 ⩾ 1, we set 𝑃(𝐼) = ∑ 𝑖∈𝐼 [𝑜, 𝑝 𝑖 ], and for later use, we extend this notation for the case 𝑘 = 0 by setting 𝑃(∅) = {𝑜}, which we regard as a 0-dimensional parallelotope. Note that for any 𝐼 = {𝑖 1 , 𝑖 2 , … , 𝑖 𝑘 } ∈  𝑍 𝑘 , the 𝑘-dimensional volume of 𝑃(𝐼) is 𝑉 𝑘 (𝑃(𝐼)), which is equal to the length of the wedge product of the vectors 𝑝 𝑖 𝑗 [15], that is, 𝑉 𝑘 (𝑃(𝐼)) = |𝑝 𝑖 1 ∧ 𝑝 𝑖 2 ∧ … ∧ 𝑝 𝑖 𝑘 |. In addition, for any parallelotope 𝑃(𝐼) with 𝐼 ∈  𝑍 𝑘 , we set 𝐵 ⟂ (𝐼) = 𝐁 𝑑 ∩ (af f 𝑃(𝐼)) ⟂ and  ⟂ (𝐼) = 𝕊 𝑑−1 ∩ (af f 𝑃(𝐼)) ⟂ , where (af f 𝑃(𝐼)) ⟂ denotes the orthogonal complement of af f 𝑃(𝐼).…”