多維圓錐上的平均數不等式

Abstract

這篇論文的主題涉及將二階錐(second order cone, SOC)上的相關定理延伸至多維圓錐(circular cone)上的類似定理。在二階錐 (SOC) 中，算術平均數、調和平均數與最大最小值有對應的大小關係，並可以形成一個不等式，本文中會將此不等式推廣至多維圓錐中，找出類似的大小關係，而在討論多維圓錐時，本文會將其分為兩個部分，即 𝜃 ∈ (0,π/4) 和 𝜃 ∈ (π/4,π/2)，進行各平均數不等式的討論，若與原不等式不相同則嘗試找出反例亦或是乘上特定的矩陣使其成立，並討論兩者之間的不同及相似之處。

The theme of this thesis involves extending some inequalities associated with second order cones (SOC) to the setting associated with circular cones.Within SOC setting, there exists a relationship among the arithmetic mean, harmonic mean, maximum values and minimum values, which can be formulated into an inequality. This thesis aims to generalize this inequality to circular cones, identifying similar relationships. When discussing circular cones, the thesis will divide them into two parts: 𝜃 ∈ (0,π/4) and 𝜃 ∈ (π/4,π/2), for the discussion of various mean inequalities. If the results differ from the original inequalities, attempts will be made to find counterexamples or multiply with specific matrices to establish its validity, followed by discussions on the differences and similarities between the two cases.