arXiv:1707.03588v4 [math.OC] 9 Sep 2018 C ftest[ set the of levels ieryidpnethprlnswt equations with hyperplanes independent linearly π bandfrom obtained oiiedfiieqartcform quadratic definite positive a nti ae esletefloigetea rbe:Gvnapositive a Given problem: extremal following subspace the affine solve sional we paper this In Introduction Notation 1.1 and Introduction 1 code: JEL M codes: AMS portfolio. financial cient Key-words: ttrsotta h ou fsltoso 111 saray a is (1.1.1) of solutions of locus the that out turns It otean space affine the to x 0 ( x ∩ stefo fteperpendicular the of foot the is Let = ) h π aineter fecetprflo neconomics. in ar portfolios Mark this efficient of of groundwork of theory mathematical treatment variance solid self-contained a lays and which meticulous subject, space a linear finite-dimensional give real we a in subspaces affine and π r r If . ( ( x r = j x ugra cdm fSine,Sfi 13 Bulgaria. 1113, Sofia Sciences, of Academy Bulgarian , ) yMroizgoer ema h nescinter fell of theory intersection the mean we geometry Markowitz By 0 = ( r π max = ) x π n ( = ) ∈ t ( x G110 ( = ] x x and ) R lisi,an usae akwt envrac hoy effi- theory, mean-variance Markowitz subspace, affine , x 0 nttt fMteaisadInformatics, and of Institute 1 56;1A9 49K35 15A99; 15A63; ∈ v n let and , τ = ) x , . . . , C,v j { , i oaiain.Let polarization). via 1 -alades [email protected] address: E-mail ( n , . . . , nMroizGeometry Markowitz On τ a C r x j 0 ) = ≤ , C n ∈ predclrt swt epc otesaa product scalar the to respect with is (perpendicularity v v etegnrcvco in vector generic the be ) v ( ⊂ R ( ̺ x ( x r } 0 j , x aetnVno Iliev Vankov Valentin 0 R ) r udaial eae along related quadratically are ) etepredclrfrom perpendicular the be = ) π fidcs n let and indices, of n ( ∈ ierform linear a , x and ) M. a 0 ̺ v then , Abstract 0 on rmteorigin the from v ( 1 R x n 0 E h min = ) n l points all find , r = π etehprln ihequation with hyperplane the be x { hc sntcntn on constant not is which ∈ ̺ C C,π r R = | n ( r x O let , ) ∩ ≥ ≥ O ftecodnt system coordinate the of j π ∈ E ( r M x M otean subspace affine the to 0 0 x } in h ) 0 , v ( eapoe subset proper a be ntepaper the In . E j C ̺ ∈ ( ) wt mean- owitz x 0 : where , ch-classical C ) hs endpoint whose = a . uhthat such = ̺ ipsoids r cr 0 n the and , 2 h . C dimen- (1.1.2) (1.1.1) ( j and , ) are (j) In case the hyperplane Π = x x1 + + xn = 1 is one of h ’s, we may interpret x C as an n-assets{ | financial··· portfolio, subject} to the linear con- straints (1.1.2).∈ Next, under certain conditions, see 4.2, we may interpret π(x) as the expected return on the portfolio x and v(x) as its risk. Finally, we may in- terpret the elements of E as efficient portfolios from Markowitz mean-variance theory in economics, considered from purely geometrical point of view. The famous pioneering work [1] is written in this fashion and the condition for non- negativity of the variables (due to lack of short sales) distorts the picture there and forces the use of variants of simplex method in Markowitz’s monograph [2]. Thus, instead of the ray E of efficient portfolios, we have to examine a more sophisticated piecewise set EM of linear segments enclosed in the compact trace ∆ of the unit simplex in Π on C. If x EM E ∆, then 0 ∈ \ ∩

π(x0) < max π(x) or v(x0) > min v(x), x∈C,v(x)≤v(x0) x∈C,π(x)≥π(x0) that is, the maximum π(x0) of the expected return decreases or the minimum v(x0) of the risk increases, which is our point of departure. In section 1, Theorem 2.2.1, we show that the trace Qa C of an ellipsoid n ∩ Qa with equation v(x) = a in R on the affine space C is again an ellipsoid in case a γM (τ), where γM (τ) is a positive definite in the ≥ M variables τ = (τj )j M R . The center of the ellipsoid Qa C is the foot of ∈ ∈ ∩ the perpendicular ̺0 from O to C, and, moreover, we find its equation in terms of appropriate coordinates on C. M The inequality a γM (τ) determines an ”elliptic” coneγ ˆM in R R , which is the base of the≥ bundle ξ described in Theorem 2.3.1. By dragging× the a = γM (τ) ”upward” (a is increasing) we establish a real algebraic variety ΓM which is the frontier ofγ ˆM and branch locus of ξ. The fibres of ξ over the points in the interior ofγ ˆM are ellipsoids which degenerate into their centers over ΓM . Using this bundle, we obtain that the image (the shadow) of an ellipsoid in Rn via projection parallel to some subspace, is again an ellipsoid — see Proposition 2.4.1. In section 2 we prove some extremal properties of the tangential points of members of a family of eccentric ellipsoids and parallel hyperplanes in Rn. These two sections stick together in section 3 where we prove that the ray E is the locus of all efficient Markowitz portfolios and give interpretation of the geometrical results in terms of Markowitz mean-variance theory.

1.2 Notation For any positive integer n we identify the members of the real linear space Rn t t with matrices of type n 1: x = (x1,...,xn), where the sign means the × t Rn n of a . We set O = (0,..., 0) and denote by (ei)i=1 the n ∈ standard basis in R . Say that M = j1,...,jm , j1 < < jm, be a proper { } ··· t subset of the set of indices [n] = 1,...,n . Given a vector x = (x1,...,xn), (M) t { } RM we denote by x the vector (xj1 ,...,xjm ) . Moreover, indexed Greek (M) t ∈ RM letters τ , etc., mean vectors (τj1 ,...,τjm ), etc., from the linear space .

2 In case K is a proper subset of the set M and we fix all τj , j K, and vary ∈ τj , j L, where L = M K, then, with some abuse of notation (the fixed components∈ are supposed to\ be known), we write τ (M) = τ (L,K). Given a symmetric n n matrix Q, by Q(M) we denote the principal m m submatrix of Q, obtained× by suppressing the rows and columns with indices× which are not in M. For a positive definite quadratic form v(x)= txQx on Rn with matrix Q we n denote Qa = x R v(x)= a , a 0. The set Qa is an ellipsoid with center n { ∈ | } ≥ O in R for all a> 0. In case n = 1 the ”ellipsoid” Qa consists of two (possibly coinciding) points. We extend this terminology by defining the singleton O to be an ”ellipsoid” when a = 0 as well as in the case of zero-dimensional linear{ } space. n n For any a 0 we denote Q a = x R v(x) a and Q) is clear. The standard scalar product (x, y)= txy in Rn produces the standard x with x 2 = (x, x). We set Sn−1 = x Rn x =1 (the unit sphere). k k k k t { ∈n |k k } The scalar product x, y = xQy in R produces the Q-norm x Q with 2 h i k k x = x, x = v(x) and the Q-distance distQ(x, y) = x y Q. Thus, the k kQ h i k − k ellipsoid Qa is a Q-sphere with Q-radius √a. Two vectors x and y are said to be Q-perpendicular, if x, y = 0. Throughout the resth of thei paper we assume that n is a positive integer and m is a nonnegative integer with m

(j) Let M [n] be a set of indices of size m, M = j1,...,jm , and let (h )j∈M be a family⊂ of linearly independent affine hyperplanes{ in }Rn. The system of coordinates can be chosen in such a way that the hyperplane h(j) has equation (M) (j) xj = τj , τj R. We denote by h(τ ) the intersection j M h . The family ∈ ∩ ∈ h(τ (M)) τ (M) RM consists of all (n m)-dimensional affine spaces in Rn, {which are| orthogonal∈ } to the m-dimensional− vector subspace generated by the vectors ej, j M. ∈ n Let Q = (qij )i,j=1 be a . For any j M we denote (Q;M c) ∈ by ρ−,j the j-th column of the (n m) n matrix obtained from Q by − × (Q;M c) deleting the rows indexed by the elements of M. Thus, ρ−,j is a vector in

3 c Rn−m (Q;M ) c (M) RM with components ρi,j = qij , i M . Given a vector τ , c ∈ c ∈ τ (M) = t(τ ,...,τ ), we set ρ(Q;M ) = m τ ρ(Q;M ). By j1 jm −,τ (M) k=1 jk −,jk P n

α(Q;M)(x)= qjkxj xk j,kX∈M we denote the quadratic form which corresponds to the principal submatrix Q(M) of Q. c (M c) Let M = i ,...,in m . In case the submatrix Q is invertible, let { 1 − } c c c x(M ) t c(Q;M ) τ (M) ,...,c(Q;M ) τ (M) = ( i1 ( ) in−m ( )) be the solution of the matrix equation

c c c Q(M )x(M ) = ρ(Q;M ). (2.1.1) − −,τ (M) We set

c (Q;M c) (M) t (Q;M ) (M) (Q;M c) (M) c (τ )= (c1 (τ ),...,cn (τ )),

c (Q;M ) (M) (Q;M c) (M) (M) where c (τ ) = τj for j M. In particular, c (τ ) h(τ ). j ∈ ∈ In case L M, L = ℓ ,...,ℓλ , we set ⊂ { 1 } c c c c(Q;M )(τ (M))= t(c(Q;M )(τ (M)),...,c(Q;M )(τ (M))). L ℓ1 ℓλ

c Note that if M = , then c(Q;[n])(τ (∅)) = 0. We write c(Q;M )(τ (M)) = c ∅ c c c(M ) τ ρ(Q;M ) ρ(M ) ( ), and, similarly, −,τ (M) = −,τ , etc., when the context allows that. c Since the vector ρ(M ) Rn−m depends linearly on τ (M), the map −,τ ∈ M n (M) (M c) (M) ψM : R R , τ c (τ ), (2.1.2) → 7→ (Q;M c) M is an injective homomorphism of linear spaces. We set Eff = ψM (R ) c and note that Eff(Q;M ) is an m-dimensional subspace of Rn. Below we use also c c the short notation Eff(M ) = Eff(Q;M ) when the matrix Q is given by default.

Lemma 2.1.1 Let K and M be proper subsets of the set of indices [n] with c c K M. Let Q(K ) and Q(M ) be invertible submatrices of Q. The following two⊂ statements are equivalent: c (i) One has c(K )(τ) h(τ (M)). c c (ii) One has c(K )(τ)=∈ c(M )(τ).

Proof: We have h(τ (M)) h(τ (K)) and let us assume K = M. It is enough c to prove that (i) implies (ii).⊂ Let c(K )(τ) h(τ (M)). We6 remind that the (j) (j) ∈ hyperplane h has equation h : xj = τj for any j M. In particular, for c c c (K ) ∈ (Kc) each j K M = M K we obtain cj (τ) = τj . Therefore c (τ)M c is a solution∈ of\ the equation\ (2.1.1). The uniqueness of this solution implies c c c(K )(τ)= c(M )(τ).

4 Corollary 2.1.2 One has

c c Eff(K ) h(τ (M)) Eff(M ) . ∩ ⊂ (M) M Now, let us fix all components of τ R , except r = τℓ for some ℓ M, so τ (M) = τ ({ℓ},M\{ℓ})(r). When we vary ∈r R, then τ ({ℓ},M\{ℓ})(r) describes∈ c a straight line in RM and hence c(M )(τ ({ℓ},M∈ \{ℓ})(r)) describes a straight line c Rn (Q;M ) (M c) ({ℓ},M\{ℓ}) in which we denote by Effℓ . Its ray c (τ (r)) r b , R (Q;M c) { | ≥ } b , is denoted by Effℓb+ . ∈ Let us set

(Q) (Q;M c) (Q;M c) γ (τ)= α Q M (τ) α Q M c (c (τ),...,c (τ)). M ( ; ) − ( ; ) i1 in−m (Q;[n]) (Q;[n]) (Q) Since α(Q;∅)(x) = 0 and c1 (τ)= = cn (τ) = 0, we obtain γ∅ (τ)= (Q) ··· 0. We write γM (τ)= γM (τ) when the matrix Q is known from the context. (M) It follows from Lemma A.1.2, (i), that γM (τ) is a quadratic form in τ . Let us move the origin of the coordinate system by the substitution x = c c z(τ (M))+ c(M )(τ (M)). Then the restrictions of the components of both x(M ) c and z(M )(τ (M)) on h(τ (M)) are coordinate functions in this (n m)-dimensional affine space. − Let v(x)= txQx be the quadratic form produced by the symmetric nonzero n n n-matrix Q. Thus, Qa : v(x) = a is a quadric in R for generic a R and × (M) (M) ∈ the real variety q (M) = Qa h(τ ) is defined in h(τ ) by the equation a,τ ∩ c t (M c) (M c) (M c) t (M ) (M c) x Q x +2 ρ x + αM (τ) a =0. (2.1.3) −,τ − Let us set

c c c c v(M )(z(τ (M))) = tz(M )(τ (M))Q(M )z(M )(τ (M)). (2.1.4)

c In case the principal submatrix Q(M ) is invertible, Lemma A.1.3 implies that (M c) (M) (M) (M) v(x) = v (z(τ )) + γM (τ ) on h , and in terms of z-coordinates the equation (2.1.3) has the form

(M c) (M) (M) v (z(τ )) = a γM (τ ). (2.1.5) − 2.2 Intersections of Ellipsoids and Affine Subspaces Let v(x) = txQx be a positive definite quadratic form produced by the sym- metric (positive definite) n n-matrix Q. This being so, Qa : v(x) = a is an n × (M c) ellipsoid in R for a> 0, Q0 = 0 , and Qa = for a< 0. In particular, Q is a principal, hence positive definite,{ } submatrice∅ of Q. Thus, the quadratic form (2.1.4) is positive definite. In accord with (2.1.3) and (2.1.5), we establish parts (ii), (iii), and (iv) of the next theorem. Part (i) is proved in Lemma A.1.2, (ii).

5 Theorem 2.2.1 Let the quadratic form v(x)= txQx be positive definite. (i) If M = , then the quadratic form γM (τ) is positive definite. 6 ∅ (ii) If a>γM (τ), then qa,τ (M) is an ellipsoid in the (n m)-dimensional (M) (M c) (M c) − h(τ ) with center c (τ) and Q -radius a γM (τ). (M c) − (iii) If a = γM (τ), then q (M) = c (τ) . p a,τ { } (iv) If a<γM (τ), then the set qa,τ (M) is empty.

Remark 2.2.2 We remind that ellipsoid in an one-dimensional affine subspace is a set consisting of two points and its center is the midpoint.

Remark 2.2.3 In accord with Lemma 3.1.2, the affine subspace h(τ (M)) is (M c) tangential to the ellipsoid Qa, a = γM (τ), at the point x = c (τ).

Remark 2.2.4 In view of the previous remark, Lemma 2.1.1 has transparent geometrical meaning: If the subspace h(τ (M)) of h(τ (K)) passes through the (Kc) (M) point x = c (τ), then h(τ ) is also tangential to Qa at x.

We obtain immediately the following corollary:

(M) Corollary 2.2.5 (i) For any x h(τ ) one has v(x) γM (τ) and an equality c holds if and only if x = c(M )(τ∈). ≥ c (ii) The point c(M )(τ) h(τ (M)) is the foot of Q-perpendicular from the origin O to the affine subspace∈ h(τ (M)) and one has

(M) (M c) distQ(O,h(τ )) = c (τ) Q = γM (τ). k k p Corollary 2.2.6 Let K and L be disjoint subsets of M with K L = M. One has ∪ (M) (i) If a = γM (τ ), then the trace qa,τ (K) of the ellipsoid Qa on the affine space h(τ (K)) is nonempty and the affine subspace h(τ (M)) h(τ (K)) is tan- (Q;M c) (M) ⊂ gential to the ellipsoid qa,τ (K) at the point c (τ ).

c c c Q(K ) M c c (ii) c(Q;M )(τ (M))= c(Q;K )(τ (K))+ c ; (τ (L) c(Q;K )(τ (K))) − L and

Kc Q( ) c (iii) γ(Q)(τ (M))= γ(Q)(τ (K))+ γ (τ (L) c(Q;K )(τ (K))). M K L − L Proof: Both assertions hold when one of the sets M, K, or L, is empty. (i) The equalities

(L) (M) (M) q (M) = q (K) h(τ )= q (K) h(τ )= Qa h(τ ) a,τ a,τ ∩ a,τ ∩ ∩ (M) and Theorem 2.2.1, (ii) – (iv), yield that under the condition a = γM (τ ) we have

(M) (Q;M c) (M) q (K) h(τ )= c (τ ) . (2.2.1) a,τ ∩ { }

6 (K) In particular, a γK (τ ) and in this case qa,τ (K) is an ellipsoid in the ≥ c (K) (K ) (K) vector space h(τ ) endowed with coordinate functions (zs (τ ))s∈Kc . The c point c(Q;K )(τ (K)) is both the origin of the coordinates and the center of the { } ellipsoid qa,τ (K) which has equation

t (Kc) (K) (Kc) (Kc) (K) (K) z (τ )Q z (τ )= a γK (τ ). − Therefore we have

(Kc) q (K) = Q (K) . a,τ a−γK (τ )

(M) (L) (K) Because of (2.2.1), the trace h(τ ) of h(τ ) on h(τ ) is tangential to qa,τ (K) (Q;M c) (M) (Q;M c) (M) at the point c (τ ) (Note that in case qa,τ (K) = c (τ ) we have (Q;M c) (M) (Q;Kc) (K) (M) { } c (τ ) = c (τ ) and h(τ ) is also tangential to qa,τ (K) at the c point c(Q;M )(τ (M)) — see Remark 3.1.1). (ii) The affine subspace h(τ (M)) is defined in h(τ (K)) by the equations c c (K ) (K) (Q;K ) (K) c zs (τ )= τs cs (τ ), s L (we have L K ). Hence the difference c c c(Q;M )(τ (M)) c−(Q;K )(τ (K)) of points∈ in the affine⊂ subspace h(τ (K)) Rn co- − (Kc) c c ⊂ Q ;M  (L) (Q;K ) (K) incides with the vector c (τ cL (τ )) and we have obtained − Kc Q( ) c (K)   (L) (Q;K ) (K) part (ii). The equalities a γK (τ ) = γL (τ cL (τ )) and (M) − − a = γM (τ yield assertion (iii).

c Q(K ) Remark 2.2.7 Since the vector c (τ (K)) is Q-perpendicular to the affine (Kc) c c (K) Q ;M  (L) (Q;K ) (K) subspace h(τ ) and since the vector c (τ cL (τ )) lies in this subspace, part (ii) of the above corollary is Pythagorean− theorem.

Remark 2.2.8 It follows from Theorem of three perpendiculars that the vec- (Kc) c c Q ;M  (L) (Q;K ) (K) tor c (τ cL (τ )) is Q-perpendicular to the affine subspace h(τ (M)). −

2.3 A Bundle Let us consider the (m + 1)-dimensional space R RM with generic vector t (M) × t (M) M (a, τ ), endowed with standard topology and letγ ˆM = (a, τ ) R R M{ ∈ × | a γM (τ) . The setγ ˆM is the closed region in R R , which consists of all ≥ } × points above the graph ΓM of the quadratic function a = γM (τ) when M = 6 ∅ andγ ˆ∅ = [0, ) 0 . In all casespra(ˆγM ) = [0, ). The set ΓM is an algebraic ∞ ×{ } M ∞ variety (hence a closed set) in R R and the differenceγ ˜M =γ ˆM ΓM is an open set, both being nonempty. × \ t (M) (M) Let γM (τ)= τ Rτ , where R is a symmetric M M-matrix. In accord with Theorem 2.2.1, (i), in case M = , the matrix R ×is positive definite. If 6 ∅ (M) M = , then R is the empty matrix. Given a 0, we set Ra = τ M ∅ (M) ≥ M { ∈ R γM (τ ) = a and note that Ra is an ellipsoid in R . Any level set | t (M) } M Γa,M = (a, τ ) R R a = γM (τ) , a > 0, is isomorphic to the { ∈ × | }

7 M (a;M c) ellipsoid Ra in R , and Γ0,M = (0, 0) . Given a 0, let us denote Eff = n (M c) t ({M) } ≥ x R x = c (τ), (a, τ ) Γa,M . We define a morphism of real algebraic{ ∈ varieties| by the rule ∈ }

n M t (M) ϕM : R R R , x (v(x), x ). → × 7→ Rn −1 Theorem 2.2.1 yields ϕM ( )=ˆγM , we set ΦM = ϕM (ˆγM ), and denote the −1 t (M) restriction of ϕM on ΦM by the same letter. Since ϕM ( (a, τ )) = qa,τ (M) , we establish the following:

Theorem 2.3.1 Let ξ = (ΦM , ϕM , γˆM ) be the bundle defined by the map ϕM . −1 t (M) (i) The restriction ξ|γ˜M is a fibration with fibres ϕM ( (a, τ )) = qa,τ (M) , t (M) n−m (M c) (a, τ ) γ˜M , which are ellipsoids in R with centers c (τ). ∈ (ii) The restriction ξ|ΓM is an isomorphism of real algebraic m-dimensional c (M ) t (M) (M c) varieties with inverse isomorphism ΓM Eff , (a, τ ) c (τ), which (a;M c)→ 7→ maps any level set Γa,M onto Eff .

(a;M c) Corollary 2.3.2 The set Eff is a real algebraic subvariety of Qa, which is isomorphic via ξ ΓM to the ellipsoid Γa,M . | Taking into account Remark 2.2.3, we obtain immediately the following:

(M) (M) Corollary 2.3.3 The family h(τ ) τ Γa,M consists of all (n m)- dimensional affine spaces in Rn{, which are| both∈ orthogonal} to the m-dimensional− vector subspace generated by the vectors ej , j M, and tangential to the ellip- ∈ soid Qa.

2.4 A Shadow M M Let us denote by ζM the restriction of the second projection pr : R R R 2 × → onγ ˆM . The composition φM = ζM ϕM is the restriction on ΦM of the pro- jection of Rn parallel to the subspace◦W defined by x(M) = 0: φ : Rn W ⊥, M c (M) −1 (M) (M) (→a;M ) φM (x)= x , and, moreover, φM (τ )= h(τ ). Since the set Eff M ⊂ Qa is mapped via φM onto the ellipsoid Ra in R and since the internal points of Qa are mapped onto the internal points of Ra, we can formulate the result from Corollary 2.3.3 as solution of a shadow problem:

Proposition 2.4.1 All (n m)-dimensional affine spaces in Rn with common − direction vector subspace W , which are also tangential to an ellipsoid Qa in n ⊥ R , intersect the orthogonal complement W at the points of an ellipsoid Ra in W ⊥ RM . All affine spaces in Rn which have nonempty intersection with the ≃ ⊥ interior of Qa and are parallel to W intersect W at the internal points of Ra.

8 3 Ellipsoids and Hyperplanes 3.1 Ellipsoids and their Tangent Spaces Let v(x) = txQx be a positive definite quadratic form. The equation of the tangent space θx of the ellipsoid Qa : v(x)= a, a> 0, at the point x Qa is 0 0 ∈

θx0 (x)= a,

t where θx (x) = x Qx. For all x Qa we have x = 0 and since the matrix Q 0 0 ∈ 6 has rank n, we obtain Qx0 = 0. In particular, θx0 is a hyperplane and Qa is a smooth hypersurface in Rn.6

Remark 3.1.1 The tangent space of the ”ellipsoid” Q0 = O at its only point n n { } x0 = O is R . In particular, any linear subspace of R is tangential to Q0.

n−1 Let a> 0 and let us fix a point x0 Qa. For any vector u S we denote n ∈ ∈ for short by Lu the line z R z = x + tu, t R . { ∈ | 0 ∈ } Lemma 3.1.2 One has

θx0 (u) θx0 (u) Lu Q a = x + tu 0 t 2 ,Lu Qa = x , x 2 u . ∩ ≤ { 0 | ≤ ≤− v(u) } ∩ { 0 0 − v(u) }

2 Proof: The inequality v(x0 + tu) a is equivalent to 2θx0 (u)t + v(u)t 0 and ≤ θ (u) ≤ the equality holds if and only if t =0 or t = 2 x0 . − v(u)

Lemma 3.1.3 Let x Qa. 0 ∈ (i) One has Q a θx ( ). ≤ ⊂ 0 ≤ (ii) One has Q a θx = Qa θx = x . ≤ ∩ 0 ∩ 0 { 0} (iii) One has Q a x θx (<). ≤ \{ 0}⊂ 0

Proof: (i) Let y Q a, y = x , and let y Lu. In accord with Lemma 3.1.2, ∈ ≤ 6 0 ∈ θx0 (u) y = x0 + tu where 0 t 2 v(u) . We have θx0 (y) = θx0 (x0)+ tθx0 (u) = ≤ 2≤ − (θx0 (u)) a + tθx (u) a 2 a. 0 ≤ − v(u) ≤ (ii) Let us suppose that there exists a point y, y = x0, with y Q≤a θx0 1 6 ∈ ∩ and let u = (y x0). Then θx (u) = 0, y Lu, and Lemma 3.1.2 implies ky−x0k − 0 ∈ Lu Q a = x — a contradiction with y Lu Q a. Now, because of the ∩ ≤ { 0} ∈ ∩ ≤ inclusions x0 Qa θx0 Q≤a θx0 = x0 , part (ii) is proved. Parts (i){ and}⊂ (ii) yield∩ part⊂ (iii).∩ { }

n We remind that hr is a hyperplane in R , defined by the equation π(x)= r, where π(x) is a non-zero linear form, and qa,r = Qa hr. ∩

Lemma 3.1.4 Let x qa,r. 0 ∈ (i) If Qa hr( ), then hr = θx . ⊂ ≤ 0 (ii) If Q

9 1 Proof: (i) When y varies through Qa x0 , then u = (y x0) varies ky−x0k n−1 \{ } − bijectively through S θx (<). On the other hand, since Qa hr( ), ∩ 0 ⊂ ≤ θx0 (u) then y Qa x0 yields π(y) r, that is, π(x0 2 v(u) u) r, and hence ∈ \{ } n−1≤ − ≤ θx0 (u)π(u) 0 for all u S θx0 (<). The last inequality also holds for all n−1 ≥ ∈ ∩ u S θx0 (>) because θx0 ( u)π( u) 0. Thus, we have θx0 (u)π(u) 0 ∈ ∩ n−1 − − ≥ n ≥ for all u S , therefore for all vectors u R . If the linear forms θx and π ∈ ∈ 0 are not proportional, then after an appropriate change of the coordinates, θx0 and π can serve as coordinate functions in Rn — a contradiction. 1 (ii) Let y Qa and let us set yn = (1 )y for any positive integer n. Then ∈ − n yn Q 0, and r > 0. The following four state- ments are equivalent: ∈ \{ } (i) One has x0 qa,r and Qx0 Rp. ∈ ∈ 2 t −1 −1 (ii) One has rQx0 = ap and a = r ( pQ p) . (iii) One has x qa,r and θx = hr. 0 ∈ 0 (iv) One has qa,r = x . { 0} Proof: (i) = (ii) Let Qx = bp, b R. We have ⇒ 0 ∈ t t t a = v(x0)= x0Qx0 = x0(bp)= b px0 = bπ(x0)= br, therefore rQx0 = ap. On the other hand, we obtain a a a2 a = tx Qx = tpQ−1 p = tpQ−1p, 0 0 r r r2 hence a = r2(tpQ−1p)−1. R t a t −1 (ii) = (i) We have Qx0 p, and, moreover, x0 = r pQ . π(x0) = t t ⇒ a t −1 a r2 ∈ t px0 = x0p = r pQ p = r a = r, hence x0 hr. Finally, v(x0) = x0Qx0 = a t −1 a t a ∈ r pQ Qx0 = r px = r π(x0)= a, therefore x0 Qa. The equivalence of parts (i) and (iii) is straightforward.∈ Part (iii) and Lemma 3.1.3, (ii), imply part (iv). (iv) = (iii) Let L = x + tz t R , z = 0, be a line in hr, that is, ⇒ { 0 | ∈ } 6 π(z) = 0. The roots of the quadratic equation v(x0 + tz) = a correspond to the intersection points of the line L and the ellipsoid Qa. Taking into account 2 that v(x0 + tz)= v(x0)+2θx0 (z)t + v(z)t , we obtain the equivalent equation 2 2θx (z)t + v(z)t = 0. Since qa,r = x , this quadratic equation has a double 0 { 0} root t = 0, that is, θx (z) = 0. Thus, we obtain L θx and therefore θx = hr. 0 ⊂ 0 0 a Corollary 3.2.2 Under conditions (i) – (iv) one has θx0 (x)= r π(x).

10 Remark 3.2.3 If x0 = 0, then parts (i), (ii), and (iv) of Lemma 3.2.1 hold for a = r = 0.

t −1 −1 (Q) 2 Let us set cp = ( pQ p) , Ep = (a, r) a = cpr , r 0 , x(a, r) = a −1 (Q) { | ≥ } Q p for any (a, r) Ep with r> 0, x(0, 0) = 0, and r ∈ Ef(Q) = x Rn x = x(a, r), (a, r) E(Q) . p { ∈ | ∈ p } Thus, the set Ef(Q) consists of all vectors x Rn which satisfy the four equiva- p ∈ lent conditions from Lemma 3.2.1. Note that 0 Ef(Q) and if x(a, r) Ef(Q), ∈ p ∈ p then x(a, r) = qa,r. In other words, Lemma 3.2.1 implies { } Corollary 3.2.4 One has

(Q) Ef = 2 qa,r. p ∪r≥0,a=cpr In case M is a singleton, Theorem 2.2.1 yields the following two corollaries:

Corollary 3.2.5 Let x, x Ef(Q), x = x(a, r), x = x(a , r ). 0 ∈ p 0 0 0 (i) If a = a , then qa,r = x . 0 0 { 0} (ii) If a>a0, then qa,r0 is an ellipsoid in the hyperplane hr0 . (iii) If a r , then qa ,r = . 0 0 ∅ Corollaries 3.2.5 and 3.2.6 imply the following two equivalent propositions:

Proposition 3.2.7 Let x, x Ef(Q), x = x(a, r), x = x(a , r ). One has 0 ∈ p 0 0 0

r0 = max r and a0 = min a. qa0,r6=∅ qa,r0 6=∅

Proposition 3.2.8 Given x Ef(Q), one has 0 ∈ p

π(x0)= max π(x) and v(x0) = min v(x). (Q) (Q) x∈Efp ,v(x)≤v(x0) x∈Efp ,π(x)≥π(x0)

(Q) It turns out that we can trow out the constraint condition x Efp from Proposition 3.2.8. We have the following theorem (compare, for∈ exam- ple, with [3, Section 2]).

Theorem 3.2.9 Let x0 qa0,r0 and r0 0. The following six statements are equivalent: ∈ ≥ (i) One has x Ef(Q). 0 ∈ p

11 (ii) One has

π(x0)= max π(x) and v(x0) = min v(x). v(x)≤a0 π(x)≥r0

(iii) One has

π(x0)= max π(x). (3.2.1) v(x)≤a0

(iv) One has

π(x0)= max π(x). v(x)=a0

(v) One has

v(x0) = min v(x). π(x)≥r0

(vi) One has

v(x0) = min v(x). π(x)=r0

Proof: Below we prove only these implications which are not straightforward. If r = 0 and x = x(a , 0) Ef(Q), then a = 0, x = 0, and the equiva- 0 0 0 ∈ p 0 0 lences hold. Now, let r0 > 0. In particular, we have x0 = 0. (i) = (ii) According to Lemma 3.2.1, (iii), and Corollary6 3.2.2 we have ⇒ a0 n x0 qa ,r and θx (x) = π(x). Let us suppose v (x) a0 for x R . Then ∈ 0 0 0 r0 ≤ ∈ Lemma 3.1.3, (i), imply π(x) r0. Now, let π (x) r0, that is, θx0 (x) a0 for some x Rn. In this case Lemma≤ 3.1.3, (iii), yields≥ v(x) a . ≥ ∈ ≥ 0 (iii) = (i) Let x satisfies condition (3.2.1). Lemma 3.1.4, (i), imply θx = ⇒ 0 0 hr0 . Now Lemma 3.2.1, (iii), finishes the proof.

(v) = (i). Since Q

4 Markowitz Geometry

In this section we unite the results from the previous two sections and give com- plete characterization of the tangent points of a family of concentric ellipsoids and a family of parallel hyperplanes in an affine subspace of Rn.

4.1 The Equality Let M = , ℓ M, and let us set L = ℓ , K = M L. Let us fix all components c of τ (K)6 ∅RK∈: τ (K) = µ(K), and set{ h}(K) = h(µ\(K)), ̺(K) = c(Q;K )(µ(K)), (K) ∈ (K) (K) ′ (M) (L,K) γ = γK (µ ). We denote r = τℓ, ̺ = ̺ℓ , r = r ̺, so τ = τ (r). (L,K) − Finally, we set a = γM (τ (r)).

12 We remind that after the translation z = x ̺(K) of the coordinate system, (Kc) − (zs)s∈Kc , where zs = zs , is a system of coordinate functions on the affine subspace h(K) with origin ̺(K). In this case h(τ (M)) = h(τ (L,K)(r)) is a hy- (K) ′ perplane in h with equation zℓ = r . In particular, the corresponding ℓ-th c coordinate vector p RK (the ℓ-th component of p is 1 and all other compo- ∈ (L,K) (K) nents are zeroes) is a normal vector of h(τ (r)) in h . We set π(x)= xℓ, (Kc) (Kc) (K) π (z)= zℓ, and note that the linear form π (z) is the restriction on h of the linear form π(x), written in terms of z. It follows from Corollary 2.2.6, (K) (i), that the trace qa,µ(K) of the ellipsoid Qa on affine space h is nonempty (L,K) and the hyperplane h(τ (r)) is tangential to the ellipsoid qa,µ(K) at the point c c(Q;M )(τ (L,K)(r)). In order to stick together notation from sections 2 and 3 in this case, we set c ′ (K) (L,K) (K ) a = a γ , h(τ (r)) = hr′ , qa′,r′ = q (K) hr′ = Q ′ hr′ . − a,µ ∩ a ∩ Theorem 4.1.1 (i) If r′ 0, then ≥ c x(a′, r′)= c(Q;M )(τ (L,K)(r)) (4.1.1) and x(0, 0) = ̺(K). (ii) One has

(Kc) (Q;M c) Q  Effℓ̺+ = Efp .

Proof: (i) The affine space h(τ (L,K)(r)) is a hyperplane in h(K), which is tan- (Q;M c) (L,K) gential to the ellipsoid qa,µ(K) at the point c (τ (r)). In particular, (Kc) c 2 ′ (Q;M ) (L,K) ′ ′ Qa′ hr = c (τ (r)) and Lemma 3.2.1, (ii), yields a = cpr for ∩t −1 {−1 } ′ ′ ′ (Q) cp = ( pQ p) . Therefore, when r 0, we have (a , r ) Ep and the ′ ≥ ′ (∈L,K) (K) equality (4.1.1) holds. In addition, if r = 0, then a = 0, γM (τ (r)) = γ , Kc Q( ) c and Corollary 2.2.6, (ii), (iii), implies γ (τ (L) c(Q;K )(µ(K))) = 0, hence L − L c Q(K ) M c c c ; ((τ (L) c(Q;K )(µ(K)))) = 0. − L In other words,

c c c(Q;M )(τ (L,K)(̺)) = c(Q;K )(µ(K)).

c This shows that x(0, 0) = c(Q;K )(µ(K))= ̺(K) and the equality (4.1.1) proves part (i) which, in turn, yields part (ii).

c c Theorem 4.1.2 Let x = c(Q;M )(τ (L,K)(r )) Eff(Q;M ). One has r = π(x ) 0 0 ∈ ℓ̺+ 0 0 and if a0 = v(x0), then

π(x0)= max π(x) and v(x0) = min v(x). (4.1.2) (K) (K) x∈h ,v(x)≤a0 x∈h ,π(x)≥r0

13 ′ Proof: According to Theorem 4.1.1, we have r0 = r0 ̺ 0, hence x0 = c Q(K ) − ≥   (K) x(a , r ) Efp . Let x = z + ̺ . Theorem 3.2.9, (i), (ii), implies 0 0 ∈ 0 0 (Kc) (Kc) π (z0)= max π (z) (K) (Kc) ′ z∈h ,v (z)≤a0 and (Kc) (Kc) v (z0) = min v (z). (K) (Kc) ′ z∈h ,π (z)≥r0

(Kc) (Kc) (K) (K) ′ Since π (x)= π(z)+̺, v(x)= v (z)+γ on h , and since r0 = r0 ̺, a′ = a γ(K), we establish the extremal property (4.1.2). − 0 0 − 4.2 The Interpretation Let k, m, and n be integers with n 2, 0 k

14 A Appendix

In this appendix we use freely notation introduced in the main body of the paper.

A.1 Three Lemmas The partition M c M = [n] of the set of indices [n] produces the following ∪ t n partitioned matrices: Any vector x = (x1,...,xn) R can be visualized as c x = t(x(M ), x(M)) and any n n-matrix Q can be visualized∈ as × c c Q(M ) Q(M ×M) c .  Q(M×M ) Q(M) 

Lemma A.1.1 Let Q be a symmetric n n-matrix and let v(x)= txQx be the corresponding quadratic form. One has ×

c c c c c v(x)= tx(M )Q(M )x(M ) +2tx(M )Q(M ×M)x(M) + tx(M)Q(M)x(M).

Proof: We have

(M c) (M c×M) t t (M c) t (M) Q Q t (M c) (M) v(x)= xQx = ( x , x ) c (x , x )=  tQ(M ×M) Q(M) 

c c c c c tx(M )Q(M )x(M ) +2tx(M )Q(M ×M)x(M) + tx(M)Q(M)x(M).

c Below we assume that Q(M ) is an .

Lemma A.1.2 Let

c c (M ) (M) (M c) −1 (M c×M) (M) (M c) (M) t (M ) (M) (M) c c (x )= (Q ) Q x , c (x )= (c c (x ), x ), M − M c c (M) t (M ) (M) (M c) (M ) (M) t (M) (M) (M) and let γM (x )= c c (x )Q c c (x )+ x Q x . − M M (M) (M) (i) γM (x ) is a quadratic form in x ,

(M) t (M) (M) t (M c×M) (M c) −1 (M c×M) (M) γM (x )= x [Q Q (Q ) Q ]x , − (M) (M c) (M) and one has γM (x )= v(c (x )). (M) (ii) If v(x) is a positive definite quadratic form in x, then γM (x ) is a positive definite quadratic form in x(M).

Proof: (i) We begin by noting that since

c c t (M ) (M) (M c) (M ) (M) t (M)t (M c×M) (M c) −1 M c×M (M) cM c (x )Q cM c (x )= x Q (Q ) Q x ,

(M) we obtain the above expression for γM (x ). On the other hand, Lemma A.1.1 implies

c v(c(M )(x(M))) =

15 c c c t (M ) (M) (M c) (M ) (M) t (M ) (M) (M c×M) (M) t (M) (M) (M) cM c (x )Q cM c (x )+2 cM c (x )Q x + x Q x . c (M c×M) (M) (M c) (M ) (M) Taking into account that Q x = Q cM c (x ), we establish the identity. − c (ii) In is enough to note that c(M )(x(M)) = 0 if and only if x(M) = 0. Now, let us translate the system of coordinates by the rule

c z(τ (M))= x c(M )(τ (M)). − Lemma A.1.3 If x(M) = τ (M), then

t (M c) (M c) (M c) (M) v(x)= z Q z + γM (τ ). Proof: In accord with Lemma A.1.1, we have

c c c c c v(x)= tx(M )Q(M )x(M ) +2tx(M )Q(M ×M)τ (M) + tτ (M)Q(M)τ (M) =

c t (M c) (M c) (M c) t (M c) (M c) (M ) (M) t (M) (M) (M) x Q x 2 x Q cM c (τ )+ τ Q τ = c − c t (M c) (M ) (M) (M c) (M c) (M ) (M) (z + cM c (τ ))Q (z + cM c (τ )) c c t (M c) (M ) (M) (M c) (M ) (M) t (M) (M) (M) 2 (z + cM c (τ ))Q cM c (τ )+ τ Q τ = − c c c t (M c) (M c) (M c) t (M ) (M c) (M ) t (M c) (M c) (M ) z Q z + cM c Q cM c +2 z Q cM c c c c − t (M c) (M c) (M ) t (M ) (M c) (M ) t (M) (M) (M) 2 z Q c c 2 c c Q c c + τ Q τ = M − M M t (M c) (M c) (M c) (M) z Q z + γM (τ ).

Asknowledgements

I would like to thank the governing body of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences for creating perfect conditions of work.

Funding

This paper was partially supported by the Bulgarian Science Fund [grand num- ber I02/18].

References

[1] H. Markowitz, Portfolio Selection, The Jornal of Finance, vol. 7, No 1, (1952), 77 - 91. [2] H. Markowitz, Portfolio Selection, 2nd ed., Blackwell 1991. [3] S. Stoyanov, S. Rachev, F. Fabozzi, Optimal Financial Portfolios, Applied Mathematical Finance, vol. 14 (2007), 401 - 436.

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