Geometry of multilinear forms on a normed space ℝ m

Abstract

UDC 514.1 For every m ≥ 2 , let ℝ ‖ ⋅ ‖ m be ℝ m with a norm ‖ ⋅ ‖ such that its unit ball has finitely many extreme points. For every n ≥ 2 , we focus our attention on the description of the sets of extreme and exposed points of the closed unit balls of ℒ ( n ℝ ‖ ⋅ ‖ m ) and ℒ s ( n ℝ ‖ ⋅ ‖ m ) , where ℒ ( n ℝ ‖ ⋅ ‖ m ) is the space of n -linear forms on ℝ ‖ ⋅ ‖ m and ℒ s ( n ℝ ‖ ⋅ ‖ m ) is the subspace of ℒ ( n ℝ ‖ ⋅ ‖ m ) formed by symmetric n -linear forms. Let ℱ = ℒ ( n ℝ ‖ ⋅ ‖ m ) or ℒ s ( n ℝ ‖ ⋅ ‖ m ) . First, we show that the number of extreme points of the unit ball of ℝ ‖ ⋅ ‖ m is greater than 2 m . By using this fact, we classify the extreme and exposed points of the closed unit ball of ℱ , respectively. It is shown that every extreme point of the closed unit ball of ℱ is exposed. We obtain the results of [Studia Sci. Math. Hungar., <strong>57</strong>, No. 3, 267–283 (2020)] and extend the results of [Acta Sci. Math. Szedged, <strong>87</strong>, No. 1-2, 233–245 (2021) and J. Korean Math., Soc., <strong>60</strong>, No. 1-2, 213–225 (2023)].