Geometry of Multilinear Forms on a Normed SpaceRm

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(ⁿR_‖·‖^m) and L_s(ⁿR_‖·‖^m), where L(ⁿR_‖·‖^m) is the space of n-linear forms on R_‖·‖^m and L_s(ⁿR_‖·‖^m) is the subspace of L(ⁿR_‖·‖^m) formed by symmetric n-linear forms. Let F=L(ⁿR_‖·‖^m) or L_s(ⁿR_‖·‖^m). First, we show that the number of extreme points of the unit ball in R_‖·‖^m is greater than 2m. By using this fact, we classify the extreme and exposed points of the closed unit ball in F, respectively. It is shown that every extreme point of the closed unit ball in F is exposed. We obtain the results of [Studia Sci. Math. Hungar., 57, No. 3, 267 (2020)] and extend the results from [Acta Sci. Math. (Szeged), 87, Nos. 1–2, 233 (2021) and J. Korean Math. Soc., 60, No. 1–2, 213 (2023)].