Geometry of multilinear forms on R^m with a certain norm

For every m ≥ 2, let R^m_k·k be R^m with a norm k · k such that | ext B_Rm_k·k| = 2m. For every n ≥ 2, we devote ourselves to the description of the sets of extreme and exposed points of the closed unit balls of L(ⁿR^m_k·k) and Ls(ⁿR^m_k·k), where L(ⁿR^m_k·k) is the space of n-linear forms on R^m_k·k, and Ls(ⁿR^m_k·k) is the subspace of L(ⁿR^m_k·k) consisting of symmetric n-linear forms. Let F = L(ⁿR^m_k·k) or Ls(ⁿR^m_k·k). First we classify the extreme and exposed points of the closed unit ball of F. We obtain || ext B_L(nRm_k·k) || = 2^(mn) and || ext B_Ls(nRm_k·k) || = 2^{dim(Ls(nRmk|·k|))}. We also show that every extreme point of the closed unit ball of F is exposed. It is shown that ext B_Ls(nRm_k·k) = ext B_L(nRm_k·k) ∩ Ls(ⁿR^m_k·k) and exp B_Ls(nRm_k·k) = exp B_L(nRm_k·k) ∩ Ls(ⁿR^m_k·k).