A note on non-commutative polytopes and polyhedra
AA NOTE ON NON-COMMUTATIVE POLYTOPES ANDPOLYHEDRA
BEATRIX HUBER AND TIM NETZER
Abstract.
It is well-known that every polyhedral cone is finitely generated(i.e. polytopal), and vice versa. Surprisingly, the two notions differ almostalways for non-commutative versions of such cones. This was obtained as abyproduct in [3] and later generalized in [8]. In this note we give a direct andconstructive proof of the statement. Our proof yields a new and surprisingquantitative result: the difference of the two notions can always be seen atthe first level of non-commutativity, i.e. for matrices of size 2, independentof dimension and complexity of the initial convex cone. This also answers anopen question from [8]. Introduction and Preliminaries
A convex cone C ⊆ R d is called polyhedral , if there exist linear functionals (cid:96) , . . . , (cid:96) m : R d → R with C = (cid:8) a ∈ R d | (cid:96) ( a ) ≥ , . . . , (cid:96) m ( a ) ≥ (cid:9) . A convex cone C is called finitely generated (or polytopal ) if there are v , . . . , v n ∈ R d with C = cc { v , . . . , v n } := (cid:40) n (cid:88) i =1 λ i v i | λ , . . . , λ n ≥ (cid:41) . The Minkowski-Weyl-Theorem (see for example [9]) states that each polyhedralcone is finitely generated, and each finitely generated cone is polyhedral.A recent development in real algebraic geometry and convexity theory is to con-sider non-commutative sets and cones. They arise by replacing points from R d with d -tuples of Hermitian matrices (of arbitrary size). A lot of meaningful infor-mation about polynomials and semialgebraic sets comes to light when these non-commutative levels are added to the classical setup. Examples are Helton’s Positiv-stellensatz [5] and the analysis of Ben-Tal and Nemirovski’s algorithm for checkinginclusion of spectrahedra [1, 3, 7], among others (see also [6] for an overview). Fora polyhedral/polytopal cone C = (cid:8) a ∈ R d | (cid:96) ( a ) ≥ , . . . , (cid:96) m ( a ) ≥ (cid:9) = cc { v , . . . , v n } there are two natural ways to extend the cone to matrix levels. The first one usesthe polyhedral description, and is the standard way of defining non-commutativesemialgebraic sets by polynomial inequalities. For each s ∈ N we define C ph s := (cid:110) ( A , . . . , A d ) ∈ Her ds | (cid:96) i ( A , . . . , A d ) (cid:62) , i = 1 , . . . , m (cid:111) , Supported by the Austrian Science Fund FWF through project P 29496-N35. a r X i v : . [ m a t h . AG ] F e b BEATRIX HUBER AND TIM NETZER where Her s is the real vector space of complex Hermitian s × s -matrices, and (cid:62) C ph1 coincides with C . We now consider the collection over all matrix-sizes as our non-commutativepolyhedral extension of C : C ph := (cid:0) C ph s (cid:1) s ∈ N . The second non-commutative extension of C uses the generators of C and looks alittle less natural at first sight. However, there are good reasons for the followingdefinition, as we will argue below. We just replace nonnegative numbers by positivesemidefinite matrices and define for any s ∈ N : C pt s := (cid:40) n (cid:88) i =1 P i ⊗ v i | P i ∈ Her s , P i (cid:62) , i = 1 , . . . , n (cid:41) . Here, ⊗ denotes the Kronecker (=tensor) product of matrices. In our case it justmeans we put P i into each component of the vector v i and multiply it with thereal number in there. The result is a d -tuple of Hermitian matrices of size s , andso is the sum over all i . Also note that C pt1 again coincides with C , since positivesemidefinite matrices of size 1 are just nonnegative real numbers. Now the collection C pt := (cid:0) C pt s (cid:1) s ∈ N is the non-commutative polytopal extension of C .We will restrict to proper convex cones from now on, i.e. closed convex cones C with nonempty interior and C ∩ − C = { } . Then all C ph s and C pt s have thesame property, and they fit well into the context of operator systems (see [3] andthe references therein for details). In fact both C ph and C pt are abstract operatorsystems with C at scalar level, and in particular convex in the non-commutativesense. It is easily seen (and shown in [3]) that C ph is the largest operator systemwith C at scalar level, and C pt is the smallest such operator system. In particularwe have C pt s ⊆ C ph s for all s (which can also be easily checked directly).2. Main Result
Theorem 1 below is the main result of these notes. Without the informationabout the matrix size 2, the result is a byproduct of the main results of [3] (seeRemark 4.9 from that work). However, the proof there is quite involved and non-constructive, in particular since the focus is on a different property of operatorsystems. See also Remark 2 below for more comments on the difference of the twoproofs. The main result was later generalized in Theorem 4.1 from [8], to includethe case that C is not polyhedral. It is also shown there that the difference ofthe cones can always be seen at level 2 d − . In Problem 4.3 the authors then askwhether this bound can be improved. We now give direct, simple and completelyconstructive proof of the main result. It also answers Problem 4.3 in proving thesomewhat surprising result about the matrix size 2. Theorem 1.
Let C ⊆ R d be a proper polyhedral cone.(i) If C is a simplex cone, then C pt = C ph .(ii) If C is not a simplex cone, then C pt2 (cid:40) C ph2 . ON-COMMUTATIVE POLYTOPES AND POLYHEDRA 3
Proof.
Statement ( i ) is easy. The argument is the same as in [3], we repeat itfor completeness. If C is a simplex cone, then up to a linear isomorphism of theunderlying space R d we can assume C = R d ≥ , the positive orthant. In that caseone readily checks C ph s = (cid:110) ( A , . . . , A d ) ∈ Her ds | A i (cid:62) , i = 1 , . . . , d (cid:111) = C pt s for all s ∈ N .For ( ii ) assume that C is not a simplex cone. We first settle the case of smallestpossible dimension, namely d = 3. Since C is proper and has at least 4 extremalrays, after applying a linear isomorphism we can assume that C is generated by v = (1 , − , , v = ( − , − , , v = ( − , , , v = (1 , , v , . . . , v n ∈ (1 , ∞ ) × ( − , × { } (see for example [4] Section 2.8.1. foran explicit construction of such an isomorphism and Figure 1 for the intersectionof the cone C with the plane defined by x = 1). x x wv v v v v v . . .v n Figure 1. section with plane x = 1 of C (blue) and D (red)We now consider the matrix tuple A := ( A , A , A ) := (cid:18)(cid:18) − (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19)(cid:19) ∈ Her and claim that A ∈ C ph2 . It is easily checked that A even fulfills A ± A (cid:62) , A ± A (cid:62) , and by Farkas Lemma [2] in particular the inequalities defining C .Let us prove A / ∈ C pt2 . First choose another point w = ( λ, − ,
1) with λ so largethat C ⊆ cc { w, v , v , v } =: D (see Figure 1). We now even prove A / ∈ D pt2 . Assume to the contrary that thereexists positive semidefinite matrices P , P , P , P ∈ Her with A = ( A , A , A )= P ⊗ w + P ⊗ v + P ⊗ v + P ⊗ v = ( λP − P − P + P , − P − P + P + P , P + P + P + P ) . BEATRIX HUBER AND TIM NETZER
Adding the first and third entry we obtain(1) (cid:18) (cid:19) = A + A = (1 + λ ) P + 2 P , which implies P = (cid:18) α
00 0 (cid:19) , P = (cid:18) α
00 0 (cid:19) for some α , α ≥
0, since P , P (cid:62) . Similarly we get (cid:18) − − (cid:19) = A − A = 2( P + P ) , implying P = 12 (cid:18) α − − (cid:19) . Plugging all of this into the equation for A we get (cid:18) − (cid:19) = (cid:18) λα − α / α / / − / (cid:19) − P , implying P = (cid:18) α / / / (cid:19) . From P , P (cid:62) α ≥ α ≥ / . From I = P + P + P + P we thus find α = α = 0, so P = P = 0, which contradicts (1). This proves A / ∈ D pt2 ⊇ C pt2 , and thus settles the case d = 3.We now proceed by induction on d . Let C ⊆ R d be a proper polyhedral conewhich is not a simplex cone. Then either C has a facet which is not a simplex cone,or a vertex figure which is not a simplex cone [10].In the first case we can assume that C is contained in the halfspace defined by x ≤ F lies in the hyperplane defined by x = 0 . Wecan apply the induction hypothesis to F ⊆ R d − and find ( A , . . . , A d ) ∈ F ph2 \ F pt2 . Then for A := (0 , A , . . . , A d ) we obviously have A ∈ C ph2 . Now assume A ∈ C pt2 .By looking at the first component in a representation(0 , A , . . . , A d ) = (cid:88) i P i ⊗ v i with v i ∈ C we see that P i (cid:54) = 0 can only occur for v i ∈ F . Indeed any v i ∈ C \ F hasa negative first entry, and such terms cannot cancel to yield 0. So the representationis a representation of ( A , . . . , A d ) in F pt2 , which does not exist. So we have shown A / ∈ C pt2 . In the second case we can assume that the non-simplex vertex-figure F of C is cut out by the hyperplane defined by x = 0, and further that v spans theonly extreme ray of C with negative x -entry, whereas all other generators have apositive first entry (see Figure 2 for an illustration). ON-COMMUTATIVE POLYTOPES AND POLYHEDRA 5 x v v v · · · v n − v n Figure 2. vertex figure F (blue) of C (red)After scaling the generators v i we can even assume that the x -component of v is −
1, and the x -component of all other v i is 1. Then the cone F is generated byvectors w , . . . , w n , where each w i is of the form w i = 12 v + 12 v i . Since F is not a simplex cone we can apply the induction hypothesis to F ⊆ R d − and again find ( A , . . . , A d ) ∈ F ph2 \ F pt2 . As before we now argue that A := (0 , A , . . . , A d ) ∈ C ph2 \ C pt2 , where A ∈ C ph2 is again clear. So assume for contradiction that A ∈ C pt2 , so thereexists some positive semidefinite P i ∈ Her with(0 , A , . . . , A d ) = P ⊗ v + P ⊗ v + · · · + P n ⊗ v n . Since the first entry of this matrix tuple is zero, we get P = P + · · · + P n , whichimplies A = ( P + · · · + P n ) ⊗ v + P ⊗ v + · · · + P n ⊗ v n = P ⊗ ( v + v ) + · · · + P n ⊗ ( v + v n )= 2 P ⊗ w + · · · + 2 P n ⊗ w n . This contradicts ( A , . . . , A n ) / ∈ F pt2 , and finishes the proof. (cid:3) Remark . (i) Let us comment on the difference of the above proof and the prooffrom [3]. First, the main result from [3] states that the abstract operator system C ph admits a finite-dimensional realization, whereas C pt does not. This of courseimplies that they cannot coincide, but gives no result on the level at which thediffer. The proof starts in a similar fashion as the above, first settling the case d = 3. But already here our construction of A is much more explicit and simplerthan what was done in [3]. The induction step in [3] is completely non-constructiveand cannot be transformed into an explicit argument. Our argument above iscompletely constructive. After applying the necessary isomorphisms and inductionsteps one obtains some explicit A ∈ C ph2 \ C pt2 . (ii) Note that all appearing matrices above are real symmetric. So the differencebetween the cones appears not only in the Hermitian case, but already when werestrict ourselves to real symmetric matrices. BEATRIX HUBER AND TIM NETZER
Example 3.
We consider the 3-dimensional square-cone C = (cid:8) a ∈ R | a ± a ≥ , a ± a ≥ (cid:9) = cc { (1 , − , , ( − , − , , ( − , , , (1 , , } . We have seen in the proof of Theorem 1 that A = (cid:18)(cid:18) − (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19)(cid:19) ∈ C ph2 \ C pt2 . So we can see the difference of the two cones for example in the affine subspace V := (cid:26)(cid:18)(cid:18) x − (cid:19) , (cid:18) yy (cid:19) , (cid:18) (cid:19)(cid:19) | x, y ∈ R (cid:27) ⊆ Her s . After identifying V with R it is a straightforward computation to see that C ph2 ∩ V = [ − , × [ − , . Determining C pt2 ∩ V needs some more computation. After imposing all necessarylinear constraints on P , P , P , P ∈ Her to ensure P ⊗ v + P ⊗ v + P ⊗ v + P ⊗ v ∈ V, then using the conditions that all P i must be positive semidefinite, and then solvingfor x and y , one gets C pt2 ∩ V = (cid:8) ( x, y ) ∈ [ − , | x + 2 y ≤ (cid:9) . Figure 3 shows the two affine sections. The black dot corresponds to the point A ∈ C ph2 \ C pt2 from above. xy A Figure 3. affine section of C ph2 (red) and C pt2 (blue) References [1] A. Ben-Tal and A. Nemirovski. On tractable approximations of uncertain linear matrix in-equalities affected by interval uncertainty.
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