Characterizations of Inner Product Spaces by Amir

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.

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