Almost locally minimal projections in finite dimensional Banach spaces (Q1293982)

Any polytopal unit ball in a \(k\)-dimensional space can, for sufficiently large \(n\), be realized as the image of the unit ball in \(\ell^n_1\) under a projection \(P\). The natural and fundamental question arises as to whether, if \(P\) satisfies the equation \(\|P\|= \lambda\) (as an operator on \(\ell^n_1\)), then the Banach-Mazur distance between \(P(\ell^n_1)\) and \(\ell^k_1\) is bounded by some function of \(\lambda\) (independent of both \(n\) and \(k\)). In previous papers [Isr. J. Math. 39, 359-364 (1981; Zbl 0501.46019) and ibid 48, 255-256 (1984; Zbl 0559.46009)] the author has shown the answer to be positive for \(\lambda\) in \([1, 1.01]\). The final section of this paper gives a possible outline for a proof of the general case. The earlier sections make progress towards such a proof by defining the notion of ``almost locally minimal projection'', giving a characterization of these projections in arbitrary finite-dimensional spaces and using that characterization to describe them more precisely on the spaces \(\ell^n_1\).