. F-?-b, j ? ? j and there exists i such that F ? ? j ? ? i

.. For-every-subset-s-?-{-1, j }, the intersection of boundaries S ?? i is included in a finite union of (n ? m)-spheres, where m is the cardinal of S

, u(W )) = max { id ? p u(W ) u(V ) , id ? p u(V ) u(W ) } . By symmetry, we only need to show that id

, By the properties of orthogonal projection

, Taking an element x ? V such that y = u(x)

?. ,

, The measure ? d,E is the unique Radon measure on G(d, E) whose total mass is 1 and which is invariant under the action of O(n) (see [Mat], for existence and unicity), Let E be an Euclidean vector space and d be a nonnegative integer

B. Lemma, Let p, q, n be non-negative integers with p + q ? n. For all Borel set A ? G(n, p + q), ? p+q, vol.2

. G(p,n)-?-q,v-?-({-w-|-v-+-w-?-a-})-dv,

, We omit W ? G(q, V ? ) for ease of notation

|. V-?-w-}, We introduce some notation so as to interpret the right-hand side as a pushforward measure

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