By Amir

Each mathematician operating in Banaeh spaee geometry or Approximation conception understands, from his personal experienee, that almost all "natural" geometrie homes could faH to carry in a generalnormed spaee until the spaee is an internal produet spaee. To reeall the weIl recognized definitions, this implies IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box gratifying: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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A maximal ellipsoid exists by compactness. If t; and t;' are two such ellipsoids we may, by affine transformation, assume that t; is the Euclidean unit sphere l: ~t = 1 and ~' is given by l: Ci2 ~t = 1. =1 n * TI i=1 (Ci2 (cl = 1. Let l: Et ~ IIxll2 since l: Ci + 1) ~t ~ IIx1l 2 , t;" be the ellipsoid a l: (Ci2 + 1) a =2. e. ~" is bounded by S. e. vol t;" > vol~, unless cl = 1 for all i. If span(S n t;) is not n-dimensional, we can stretch ~ in the orthogonal direction and get a larger volume. 1, and let 11,11, I· I be the corresponding norms.

Let S be the unit sphere of a norm on IRn. Then there are a unique ellipsoid ~o of maximal volume bounded by Sand a unique ellipsoid ~1 of minimal volume bounding S (these are "Loewner ellipsoids" of S). In each case, dirn span(S n ~i) = n. Proof: By analogy (or, by duality), it suffices to treat t;o. A maximal ellipsoid exists by compactness. If t; and t;' are two such ellipsoids we may, by affine transformation, assume that t; is the Euclidean unit sphere l: ~t = 1 and ~' is given by l: Ci2 ~t = 1.

0 E PLx, Y E L ==> x # y). , every proxirninal subspace is centrally symmetrie). For many of the conditions, the validity of the condition in a dense subset of SE or, respectively, of SE x SE, implies automatically its validity in all of SE or SE x SE. g. v]O whenever u,v are smooth. : and Z E p[u,y]O (u + v). is smooth in span(u,v). 4) immediately implies strict convexity. v]O. 'II]O. 4) is violated. If z is not smooth in F, approach z from the left and from the right, respectively, by smooth z~, z"n E IIzll SF.