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Russian Academy of SciencesSt. Petersburg division of Steklov Mathematical InstituteEuler overseas Mathematical InstituteInternational convention Polynomial desktop AlgebraDedicated to the reminiscence of E.V. Pankratiev (1945-2008)April 4-7, 2008

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LXi)\H(Xi, (Xi) n = lXi) i=1 n i= 1 which implies (a). 54 2. Convex Polytopes (b) Let F be a facet of Q. Let x be a relative interior point of F. Then F is the smallest face of Q containing x, cf. 6. ) n Q, cf. 5. (c) For n = 1 there is nothing to prove. So assume that n > 1. j) n Q is a proper face of Q. j) n Q is a facet. ) n int ;= 1 ;*i cf. 1. ), the desired conclusion follows. ;) n ;= 1 i#:j for some j. j) n Q is a facet of Q. j) n Q. f. ;) n Q. ;) n Q, cf. 5. ;), a contradiction. ) n Q is not a facet of Q.

N Ci)O = clconv U Ci ieI lEI when the sets C j are closed convex sets containing o. 5. • , x d) E [Rd such that x e + I = ... = Xd = O. Let IT denote the orthogonal projection of [Rd onto [Re. Show that for any subset M of [Rd we have IT(M)" MO n = where IT(M)O denotes the polar of IT(M) in [Re [Re, and MO denotes the polar of M in [Rd. 6. Let C and D be mutually polar compact convex sets. Let F be a proper exposed face of C, and let G := F6. Show that G= Dn n H(x, 1), XE extF and show that G = D n H(xo, 1) for any relative interior point Xo of F.

The image of a polyhedral set under an affine mapping is again polyhedral. The facial structure of a (non-empty) polyhedral set Q in [Rd is trivial when Q is an affine subspace of [Rd, the only faces being 0 and Q. When Q is an e-dimensional polyhedral set in [Rd which is not an affine subspace, then Q is affinely isomorphic to a polyhedral set Q' in [Re with dim Q/ = e and Q/ # [Re. Therefore, when studying facial properties of polyhedral sets, it suffices to consider polyhedral sets Q in [Rd with dim Q = d and Q # [Rd.

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