Subspaces of small codimension of finite-dimensional Banach spaces

Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking values in a Hilbert space, we deduce an inequality for entropy numbers of the operator and its dual. In this note we study the following problem: given an n-dimensional Banach space E and a Euclidean norm 1j j 112 on E and 0 An such that (*) 11x12 < M*f(1 A)IIxlI for x E F. Here M* denotes the Levy mean of the dual norm of E (see the notation below). This problem was considered by V. Milman, who proved in [18] that estimate (*) holds for a certain exponential function f. The estimate was improved later in [10] to f(1 A) < K/(1 A), where K is a universal constant. The latter result turned out to be important for various applications (cf. [1, 15, 11, 19]). The main result of this note proves (*) with the function f(1 A) < K/v'?. This estimate, besides being optimal (up to a logarithmic factor), can be used to compare entropy numbers of an operator and its dual for operators taking values in a Hilbert space. Let us recall some notation. Let E be an n-dimensional Banach space; i.e., E = (Rn, * 11). Let [.,] be an inner product on Rn, and let j . j be the associated Euclidean norm on Rn defined by I11xI11 = [x,x]1/2, for x E Rn. Let BE be the closed unit ball in E. Set (1) gIIxII*=sup{I[x,y]IIyEBE} forxERn. Clearly, (Rn, jj *I1) can be identified with the dual space E*. Let S = {x E Rnl II xjII = 1}, and let ti be the normalized rotation invariant measure on S. Define the Levy means M and M* by M= (If I1xl2d}) , M,= (j !1XI2Idu) We shall employ a similar notation in a context of symmetric convex bodies. For a closed symmetric convex body V c Rn, by V* we denote the dual body defined byV* = {x Rnl [X,y]I < 1 forallyEV}. Received by the editors May 29, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20; Secondary 47B10. lResearch partially supported by NSERC Grant A8854. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page