A Spectral Approach to Polytope Diameter
AA Spectral Approach to Polytope Diameter
Hariharan Narayanan ∗ TIFR Mumbai Rikhav ShahUC Berkeley Nikhil Srivastava † UC BerkeleyMarch 4, 2021
Abstract
We prove upper bounds on the graph diameters of polytopes in two settings. Thefirst is a worst-case bound for integer polytopes in terms of the length of the descriptionof the polytope (in bits) and the minimum angle between facets of its polar. The secondis a smoothed analysis bound: given an appropriately normalized polytope, we addsmall Gaussian noise to each constraint. We consider a natural geometric measure onthe vertices of the perturbed polytope (corresponding to the mean curvature measureof its polar) and show that with high probability there exists a “giant component” ofvertices, with measure 1 − o (1) and polynomial diameter. Both bounds rely on spectralgaps — of a certain Schr¨odinger operator in the first case, and a certain continuoustime Markov chain in the second — which arise from the log-concavity of the volume ofa simple polytope in terms of its slack variables. Contents π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Average Jump Rate Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Proof of Lemma 4.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Proof of Lemma 4.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Removing Assumptions (S),(R) . . . . . . . . . . . . . . . . . . . . . . . . . 23 ∗ Supported by a Swarna Jayanti fellowship † Supported by NSF Grants CCF-1553751 and CCF-2009011. a r X i v : . [ m a t h . C O ] M a r Proof of Theorem 2.2 27
The polynomial Hirsch conjecture asks whether the diameter of an arbitrary bounded polytope P = { x ∈ R d : Ax ≤ b } is at most a fixed polynomial in m and d . This conjecture is widelyopen, with the best known upper bounds being ( m − d ) log d − log log d + O (1) ([Suk19], see also[KK92, Tod14]) and O ( m ) for fixed d ([Lar70, Bar74]); the best known lower bound is (1+ (cid:15) ) m for some (cid:15) > d is sufficiently large [San12]. Given this situation, there has beeninterest in the following potentially easier questions:Q1. Assuming A, b have integer entries, bound the diameter of P in terms of their size.Q2. Assuming A, b are sampled randomly from some distribution, bound the diameter of P with high probability.Progress on these questions ([BDSE +
14, BR13, EV17, DH16], [Bor12, ST04, Ver09, DH20a])has relied mostly on techniques from polyhedral combinatorics, integral geometry, probability,and operations research (e.g., analysis of the simplex algorithm and its cousins).On the other hand, the Brunn-Minkowski theory of polytopes has developed largely sepa-rately over the past century, with several celebrated achievements including the Alexandrov-Fenchel inequality [Ale37] and more generally the Hodge-Riemann relations for certainalgebras associated with simple polytopes [Tim99]. One consequence of this theory is that acertain Schr¨odinger operator (weighted adjacency matrix plus diagonal) associated with thegraph of every bounded polytope has a spectral gap [Izm10] (see Definition 2.1 and Theorem2.2). We use this fact to make progress on Q1 and Q2. In the first setting, we show thefollowing theorem.
Theorem 1.1.
Suppose P = { x ∈ R d : Ax ≤ b } is a bounded polytope with integer coefficients A ∈ Z m × d , b ∈ Z m such that every minor of A, b has determinant bounded by ∆ . Then P hasdiameter O ( d ∆ log( m ∆)) . Theorem 1.1 follows from a more geometric result (Theorem 3.2) stated in terms ofthe angles between the facets of the polar of P , which is proven in Section 3. It comparesfavorably (in the regime m (cid:28) d ) with the best previously known result of this kind due to[DH16], who achieved a bound of O ( d ∆ ). However, our diameter bound is nonconstructivewhereas [DH16] show how to efficiently find a path between any two vertices of P ; we referthe reader to the introduction of that paper for a more thorough discussion of previous workin this vein (originally initiated by [DF94, BDSE + d in comparison with previous works is that they rely on combinatorialexpansion arguments, whereas we use spectral expansion, which is amenable to a “squareroot” improvement using Chebyshev polynomials.Regarding Q2, the study of diameters of random polytopes began with the influentialwork of Borgwardt [Bor77, Bor12], who considered A with i.i.d. standard Gaussian entries2nd b = 1. Borgwardt showed the following “for each” guarantee: for any fixed objectivefunctions c, c (cid:48) ∈ R d , the combinatorial distance between the vertices x, x (cid:48) of P maximizing (cid:104) c, x, (cid:105) , (cid:104) c (cid:48) , x (cid:48) (cid:105) is at most O ( d / √ log m ) in expectation, provided m → ∞ sufficiently rapidly.This type of result was extended to the “smoothed unit LP” model by Spielman and Teng inthe seminal work [ST04]; in this model one takes P = { x ∈ R d : (cid:104) x, v j (cid:105) ≤ } (1)where v j ∼ N ( a j , σ ) for some fixed vectors a , . . . , a m normalized to have (cid:107) a j (cid:107) ≤
1. Theoriginal poly( m, d, σ − ) diameter bound of [ST04] was improved and simplified in [DS05,Ver09, DH20a]; a key ingredient in each of these results was a “shadow vertex bound”analyzing the expected number of vertices of a two-dimensional projection of P . Note thatall of these results provide “for each” guarantees: at best they bound the distance between asingle pair of vertices, not between all pairs.Our second contribution is to prove that for the smoothed unit LP model, most pairs ofvertices in P are polynomially (in m, d, σ − ) close with high probability, where most is definedwith respect to a certain locally defined measure on the vertices known as the mean curvaturemeasure χ in convex geometry (see [Sch14, Sch94]; we recall the definition in Section 4). Inthe language of random graph theory, this means that the graph of P likely contains a “giantcomponent” with respect to χ which is of small diameter. Theorem 1.2.
Assume P is a random polytope sampled from the smoothed LP model. Let χ denote the mean curvature measure on the facets of P ◦ , which corresponds naturally to ameasure on the set of vertices of P , denoted Ω . Then with probability at least − / poly( m ) ,for every ψ > there is a subset G := G ( ψ ) ⊂ Ω with χ ( G ) ≥ (1 − ψ ) χ (Ω) such that thevertex diameter of G is at most O (cid:18) poly( m, d ) σ ψ (cid:19) . (2)We prove Theorem 1.2 in Section 4.5, where we deduce it from a more refined theorem(Theorem 4.1, which includes explicit powers of m, d ) for a certain class of well-roundedpolytopes. The idea of the proof is to consider a certain continuous time Markov chain whosestates are the vertices of P . This chain automatically has a large spectral gap by Theorem 2.2and the main challenge is to bound its average transition rate. This is carried out in Sections4.2-4.4 and involves further use of the Alexandrov-Fenchel inequalities, tools from integralgeometry, Gaussian anticoncentration, and an application of the shadow vertex bound of[DH20a]. Remark 1.3 (Expansion of Polytopes) . There has been a sustained interest in studying theexpansion of graphs of combinatorial polytopes beginning with [Mih92] which conjecturedthat all 0 / / Preliminaries and Notation.
We recall some basic terminology and facts regardingpolytopes; the reader may consult [Sch14, Chapter 4] for a more thorough introduction.We denote the convex hull of a set of points by conv( · ) and its affine hull by aff( · ). Let P = { x ∈ R d : Ax ≤ b } with A ∈ R m × d , b ∈ R m> be a bounded polytope containing the originin its interior. Its polar is the polytope P ◦ = conv { b − j a j } mj =1 =: K, where a T , . . . , a Tm are the rows of A .A polytope in R d is called simple if each of its vertices is contained in exactly d codimension-1 facets, and simplicial if each codimension-1 facet contains exactly d vertices. Unless otherwisenoted, “facet” refers to a codimension-1 facet. The polar of a simple polytope is simplicialand vice versa.The 1 − dimensional facets of a polytope are called edges , and are all line segments whenit is bounded. The vertex diameter of a bounded polytope P is the diameter of the graph ofits vertices and edges. Two ( d − adjacent if their intersection isa ( d − facet diameter of a polytope K is the diameter of thegraph with vertices given by its facets and edges given by the adjacency relation on facets.By duality, the vertex diameter of a simple polytope P is equal to the facet diameter of P ◦ .We use dist( · , · ) to denote the Euclidean distance between two subsets of R d , andhdist( L, K ) := max (cid:26) sup x ∈ L dist( x, K ) , sup y ∈ K dist( y, L ) (cid:27) to denote the Hausdorff distance between two sets.We use V ( K [ j ] , L [ d − j ]) to denote the mixed volume of j copies of K and d − j copiesof L for convex bodies K, L ⊂ R d . The Alexandrov-Fenchel inequalities imply that theseare log-concave, in the sense that for j , j , j = βj + (1 − β ) j integers in { , . . . , d } with β ∈ [0 ,
1] then V ( K [ j ] , L [ d − j ]) ≥ V ( K [ j ] , L [ d − j ]) β · V ( K [ j ] , L [ d − j ]) − β . (3)We use C to denote absolute constants whose value may change from line to line, unlessspecified otherwise. In this section, we recall that a certain matrix associated with every bounded polytope hasexactly one positive eigenvalue. 4 efinition 2.1 (Formal Hessian) . For K a bounded polytope containing the origin in itsinterior with N facets labeled { , . . . , N } , let H ( K ) denote the N × N matrix with entries ( H ( K )) ij = (cid:40) | F ij | csc( θ ij ) i (cid:54) = j − (cid:80) k | F ik | cot( θ ik ) i = j (4) where F ij is the intersection of facets i and j , and θ ij is the angle between the vectors normalto those faces, facing away from the origin. When K is simple, H ( K ) is the Hessian of the volume of K ( c ) = { x | M x ≤ c } withrespect to the slack vector c >
0. (see [Sch14, Chapter 4]). Log-concavity of the volumeimplies that this Hessian has exactly one positive eigenvalue. Izmestiev [Izm10] has shownvia an approximation argument that this remains true for the formal Hessian of any polytope.
Theorem 2.2 (Theorem 2.4 of [Izm10]) . H ( K ) has exactly one positive eigenvalue for anybounded polytope K . We include a self-contained proof of Theorem 2.2 in Appendix A for completeness .We will apply Theorem 2.2 to certain matrices derived from the formal Hessian and thefollowing diagonal scaling, which plays an important role in the remainder of the paper. Definition 2.3.
Let
K, F ij , θ ij be as in Definition 2.1. Then let D ( K ) denote the N × N positive diagonal matrix with entries ( D ( K )) ii = (cid:80) k F ik tan( θ ik / . Note that θ ik (cid:54) = π whenever F ik = 0 since parallel facets of a convex polytope cannot intersect. Lemma 2.4 (Spectral Gaps from Log-Concavity) . Let K be a polytope and take H := H ( K ) , D := D ( K ) . Let L be the graph Laplacian with entries: L ij = (cid:40) − F ij csc( θ ij ) i (cid:54) = j (cid:80) k F ik csc( θ ij ) i = j . (5) Then1. D − / HD − / has exactly one eigenvalue at with the rest of the eigenvalues in ( −∞ , .The eigenvector corresponding to this eigenvalue is D / .2. − D − L has exactly one eigenvalue at zero, with the rest of the eigenvalues in ( −∞ , − .The left corresponding to this eigenvalue is D .Proof. Observe that H is “nearly” a graph Laplacian in the sense that: H = − L + D (6)where we have used the identity − cos θ sin θ = tan( θ/ H matches that of D − / HD − / = − D − / LD − / + I, (7) Our proof yields a slightly stronger conclusion regarding continuity of the formal Hessian than [Izm10]. L (cid:23) L = 0, so by Sylvester’s law − D − / LD − / (cid:22) D − / HD − / has exactly one eigenvalue equal to one, with eigenvector D / and the rest of the eigenvalues nonpositive, establishing the first claim. The second claimfollows from (7) and the similarity of D − L and D − / LD − / . In this section we use the spectral gap bound of Lemma 2.4(1) to give a bound on the diameterof a polytope specified by integer constraints. We begin by generalizing the argument of[VDH95], who used Chebyshev polynomials to control the diameter of regular and biregulargraphs in terms of their spectra, to handle the matrix D − / HD − / by appropriatelycontrolling its negative entries and top eigenvector. Lemma 3.1 (Diameter in terms of Spectrum) . Let A be a weighted real symmetric adjacencymatrix (possibly with self-loops and negative weights) for a graph G on N vertices. Supposefor some constant g > there is exactly one eigenvalue of A at g with correspondingeigenvector v , the smallest absolute entry of which is v min . Further suppose that the rest ofthe eigenvalues of A are in the interval [ − , . Then the diameter of G is at most log( N ) + 2 log | v min | − √ g . Proof.
Note that if M ∈ span( I, A, · · · , A pk ) then e Ti M e j (cid:54) = 0 if and only if there is a pathin G from i to j of length at most pk . To this end, consider T pk ( A ) where T k is the degree k Chebyshev Polynomial of the first kind. If we find that T pk ( A ) (cid:54) = 0 entry-wise, then we canconclude the diameter of G is at most pk . Let A = vv T (1 + g ) + N (cid:88) i =2 u i u Ti λ i be the spectral decomposition of A . Let | · | denote the entry-wise absolute value. Then | T pk ( A ) − vv T T pk (1 + g ) | = | N (cid:88) i =2 u i u Ti T pk ( λ i ) | ≤ N (cid:88) i =2 | u i u Ti | | T pk ( λ i ) | ≤ N (cid:88) i =2 | u i u Ti | ≤ N. We would therefore have T pk ( A ) (cid:54) = 0 entry-wise if N is smaller then the smallest absoluteentry of vv T T pk (1 + g ), which is lower bounded by v T pk (1 + g ) ≥ v (1 + 0 . gk ) p . Itsuffices to pick k = 1 . / √ g and p = log ( N/v ). Theorem 3.2.
Let P = { x ∈ R d : Ax ≤ b } be a bounded polytope containing the origin,defined by m integer constraints in d dimensions with m ≥ d . Assume all angles between airs of adjacent facets of P o are at least θ . Let the bit-length of each entry of A, b be atmost B . Then the vertex diameter of P is O (cid:18) d log m + dB sin( θ ) (cid:19) . Proof.
Put D := D ( P o ) and H := H ( P o ). By Lemma 2.4, D − / HD − / is real symmetricwith one eigenvalue at 1 and the rest at most 0. We can bound its smallest eigenvalue byusing Lemma 3.3 and considering the similar matrix D − H . We upper bound the absoluterow sum of the i th row of D − H by (cid:88) j | ( D − H ) ij | ≤ (cid:80) i ∼ j F ij csc( θ ij ) (cid:80) i ∼ k F ik tan( θ ik /
2) + (cid:80) i ∼ j F ij | cot( θ ij ) | (cid:80) i ∼ k F ik tan( θ ik / ≤ (cid:80) i ∼ j F ij csc( θ ij ) (cid:80) i ∼ k F ik tan( θ ik / . Taking the supremum of the above expression givessup i (cid:80) k F ik csc( θ ik ) (cid:80) k F ik tan( θ ik / ≤ sup i ∼ j θ ij )tan( θ ij /
2) = sup i ∼ j csc ( θ ij / . Therefore by Lemma 3.3, the smallest eigenvalue of D − H , and consequently of D − / HD − / is at least − csc ( θ / M = D − / HD − / + csc ( θ / ( θ / ( θ / ( θ / = 1 + csc − ( θ /
2) with the rest contained in theinterval [0 , g = csc − ( θ /
2) to obtain a diameter of O (cid:18) log N + log | v min | − sin( θ / (cid:19) . The eigenvector v corresponding to eigenvalue 1 + csc ( θ /
2) is simply T D / normalized, so v min = min i ( T √ D ) i || T √ D || ≥ √ N (cid:114) min i D ii max i D ii ≥ √ N min i (cid:80) k F ik tan( θ ik / i (cid:80) k F ik tan( θ ik / ≥ θ θ N / min i,j F ij max i,j F ij , where θ > θ ik ≤ π − θ for all i ∼ k . Finally use N ≤ m d ,the approximation sin( θ / ≥ C sin( θ ), as well as θ θ ≥ − O ( d log d + dB ) and max i,j F ij min i,j F ij ≤ O ( dB + d log d ) from Lemma 3.4 for the end result. Lemma 3.3 (Gershgorin’s circle theorem) . The smallest (real) eigenvalue of M is at least − sup i (cid:80) j | M ij | . Lemma 3.4 (Worst Case Volumes and Angles) . Let P o = conv( a /b , · · · , a m /b m ) be apolytope where each a i /b i ∈ R d is a vertex and a i ∈ R d , b i ∈ R have integer entries withabsolute value at most B . Then: . The smallest co-dimension facet of P o has volume at least − O ( dB + d log d ) , and thelargest co-dimension facet has volume at most dB + d log d .2. The angle between any two adjacent facets is contained in [2 − O ( d log d + dB ) , π − − O ( d log d + dB ) ] .3. If the largest × and ( d − × ( d − minors of A are bounded in magnitude by ∆ and ∆ d − respectively, then csc( θ ) = O ( d ∆ ∆ d − ) Proof.
Every co-dimension 2 facet can be written as the convex hull of some subset ofsize at least d − a /b , · · · , a m /b m . Without loss of generality, say that F = conv( a /b , · · · , a d − /b d − ) is the smallest co-dimension 2 facet. Then its volume is:Vol(conv( a /b , · · · , a d − /b d − )) = 1( d − (cid:112) | det( M T M ) | ≥ d !( b . . . b d − ) b dd − ≥ − O ( d log d + dB ) , where M is the d × ( d −
2) matrix whose i th column is a i /b i − a d − /b d − , and we have usedthat the determinant of a nonsingular integer matrix is at least one. On the other hand, P o is contained inside the (cid:96) ball of radius d / B ≤ B +log d , and so each co-dimension 2 facet of P o is contained in a cross section of that ball, so has volume at most (2 B +log d ) d ≤ dB + d log d ,establishing (1).Regarding the angles, consider without loss of generality a facet F = conv( a /b , . . . , a d /b d )and a vertex a j , j > d of a facet adjacent to F . Observe that the angle θ between the normalsto these adjacent facets satisfies: csc( θ ) = dist( a j , F )dist( a j , aff( F )) . The numerator is at most the distance between two vertices of P ◦ , which is at mostmax i,j (cid:107) a i /b i − a j /b j (cid:107) ≤ √ d ∆ . The denominator is given by dist( a j /b j , aff( F ))where M is the d × d matrix with columns a j /b j − a /b , a /b − a /b . . . , a d /b d − a /b ,which must be invertible since conv( a /b , . . . , a d /b d , a j /b j ) is a full dimensional simplex. Bythe adjugate formula, the entries of M − are of magnitude at most ∆ d − , so we havedist( a j , aff( F )) ≥ √ d ∆ d − . Combining these bounds yields csc( θ ) = O ( d ∆ ∆ d − ) , establishing (3).To obtain (2), observe that that ∆ ≤ B and ∆ d − ≤ ( d − dB ≤ O ( d log d )+ dB .8inally, we can prove the bound advertised in the introduction. Proof of Theorem 1.1.
Applying Theorem 3.2, Lemma 3.4(3), and the relation ∆ ≤ B , wefind that the diameter of P is at most O ( d ∆ ∆ d − (log m + log(∆ )) , which is at most the desired bound. In this section we consider the “smoothed unit LP” model defined in (1). Suppose P is afixed polytope specified as P = { x ∈ R d : (cid:104) a j , x (cid:105) ≤ , j ∈ [ m ] } , for some vectors (cid:107) a j (cid:107) ≤
1, and consider the random polytope P = { x ∈ R d : (cid:104) v j , x (cid:105) ≤ } , where v j = a j + g j for g j ∼ N (0 , σ I d ) i.i.d spherical Gaussians. Denote the polars of P and P by K := P ◦ = conv( a , . . . , a m ) ⊂ B d ,K := P ◦ = conv( v , . . . , v m ) . Note that K is simplicial with probability one, so each of its k -dimensional facets hasexactly k + 1 vertices. We will use the notation F S := conv { v j : j ∈ S } to denote facetsof K and F k ( K ) := (cid:110) S ∈ (cid:0) [ m ] k +1 (cid:1) : F S is a k -dimensional facet of K (cid:111) to denote the set of allfacets of K . The k -dimensional volume of a facet F S , S ∈ F k ( K ) will be denoted by | F S | orVol k ( F S ). We will often abbreviate F S ∩ T as F ST for adjacent S, T . For two
S, T ∈ (cid:0) [ m ] d (cid:1) , let θ ST ∈ (0 , π ) denote the angle between the unit normals u S , u T to F S , F T , respectively; notethat almost surely θ ST (cid:54) = 0 , π for every S, T ∈ (cid:0) [ m ] d (cid:1) .We will pay special attention to the set of ( d − K , which we denote asΩ := F d − ( K ) ⊂ (cid:18) [ m ] d (cid:19) . Define the measures χ , π, δ : Ω → R ≥ by χ ( S ) := (cid:88) T ∈ Ω | F ST | θ ST , (8) π ( S ) := (cid:88) T ∈ Ω | F ST | tan( θ ST / , (9)9 ( S ) := (cid:88) T ∈ Ω | F ST | csc( θ ST ) . (10)It will be convenient to make two further technical assumptions on K and σ for theproofs of our results; in Section 4.5 we will show that any instance of the smoothed unit LPmodel may be reduced to one satisfying both assumptions with parameter r = Ω( σm ) , (11)incurring only a poly( m ) loss in the diameter. Let K ( j )0 = conv( a i : i (cid:54) = j ) be the polytopeobtained from K by deleting vertex a j . (R) Roundedness of Subpolytopes: There is an r ∈ (0 ,
1) such that for every j ≤ m : rB d ⊂ K ( j )0 . (S) Smallness of σ : α := 6 σ (cid:112) d log m < r/d . (12)The main result of this section is the following “almost-diameter” bound with respect tothe measure π . Theorem 4.1.
Assume ( S ) , ( R ) . Then with probability at least − / log( m ) , for every φ > there is a subset G := G ( φ ) ⊂ Ω with π ( G ) ≥ (1 − φ ) π (Ω) such that the facet diameterof G is at most ˜ O ( m d /σ rφ ) . Remark 4.2.
The probability in Theorem 4.1 may be upgraded to 1 − m − c for any c at thecost of an additional m c factor in the diameter bound, by applying Markov’s inequality inthe proof of Lemma 4.5 with a different threshold.The proof of Theorem 4.1 relies on three properties of the (random) continuous timeMarkov chain with state space Ω and infinitesimal generator Q := − D − L, (13)where L is as in (5). The corresponding Markov semigroup P ( t ) := exp( − tD − L ) , t ≥ , has stationary distribution proportional to D = π ( · ) by Lemma 2.4(2); call the normalizedstationary distribution π ( · ) := π ( · ) /π (Ω)The first property is that the stationary distribution π is (in a quite mild sense) non-degenerate, with high probability. Apart from being essential in our proofs, this relates themeasure π to well-studied measures in convex geometry such as the surface measure andmean curvature measure χ ( · ), clarifying the meaning of Theorem 4.1. The proof of Lemma4.3 appears in Section 4.1. The reader may consult e.g. [G +
20, Chapter 6] for an introduction to continuous time Markov processes. emma 4.3 (Non-degeneracy of π ) . Assume ( S ) , ( R ) . With probability at least − /m :1. min S ∈ Ω π ( S ) ≥ π min := C m − d r d . c Vol d − ( ∂K ) ≤ π (Ω) ≤ O ( d r − )Vol d − ( ∂K ) .
3. For every S ∈ Ω , χ ( S ) / ≤ π ( S ) ≤ O ( r − ) χ ( S ) . The second property is that Q (almost surely) has a spectral gap of at least one, byLemma 2.4(2). This implies that the chain (13) mixes rapidly to π (in the sense of continuoustime) from any well-behaved starting distribution. In particular let us say that a probabilitymeasure p on Ω is an M -warm start ifsup S ∈ Ω p ( S ) π ( S ) ≤ M. Let (cid:96) ( π ) denote the inner product space on defined on R Ω , where the inner product is givenby (cid:104) f, g (cid:105) (cid:96) ( π ) := (cid:80) S ∈ Ω π ( S ) f ( S ) g ( S ), and let (cid:96) ( π ) be the corresponding (cid:96) space. Let Π bethe Ω × Ω diagonal matrix whose S th diagonal entry is π ( S ). We define the density of p withrespect to π to be the the vector with entries p ( S ) π ( S ) . Lemma 4.4 (Warm Start Mixing) . If p is M − warm, then for τ > , t = Ω(log( M/τ )) time,one has || π − pP ( t ) || T V ≤ τ. Proof.
By Lemma 2.4(2), we see that the top eigenvalue of exp( − tD − L ) is 1, and all theothers are positive but less or equal to e − t . The (cid:96) ( π ) norm of any density ρ of an M − warmdistribution p is at most M . It follows that || − ρ Π P ( t )Π − ) || (cid:96) ( π ) ≤ τ, for t ≥ ln Mτ . However, (cid:107) ( − pP ( t )Π − ) (cid:107) (cid:96) ( π ) is at least as large as (cid:107) ( − pP ( t )Π − ) (cid:107) (cid:96) ( π ) ,which equals || π − pP ( t ) || T V . The lemma follows.The third and final property is a bound on the rate at which the continuous chain makesdiscrete transitions between states. Let J avg denote the average number of state transitionsmade by the continuous time chain in unit time, from stationarity, and note that J avg = (cid:80) S ∈ Ω π ( S ) | Q ( S, S ) | (cid:80) S ∈ Ω π ( S ) = (cid:80) S ∈ Ω δ ( S ) (cid:80) S ∈ Ω π ( S )as the diagonal entries of the generator Q are equal to − δ ( S ) /π ( S ). The most technical partof the proof is the following probabilistic bound.11 emma 4.5 (Polynomial Jump Rate) . Assume ( S ) , ( R ) . With probability at least − / log( m ) , the continuous time Markov chain defined by (13) satisfies: J avg ≤ ˜ O ( m d /σ r ) . The proof of this lemma involves showing that the facets of K are well-shaped and havenon-degenerate angles between them in a certain average sense, and is carried out in Sections4.2, 4.3, and 4.4.Combining these ingredients, we can prove Theorem 4.1 Proof of Theorem 4.1.
Let T be a fixed positive time to be chosen later. Consider thecontinuous time chain (13), and for F ∈ Ω let the random variable J TF denote the number oftransitions in [0 , T ] when the chain is started at F . With probability 1 − /m we have (cid:88) F ∈ Ω π ( F ) E J TF = T J avg ≤ ˜ O ( m d /σ r ) · T by Lemma 4.5 so there is a facet F ∈ Ω satisfying E J TF ≤ ˜ O ( m d /σ r ) · T. (14)By Lemma 4.3(1), the distribution δ F concentrated on F is π − min − warm with probability1 − /m . Invoking Lemma 4.4 with starting distribution δ F and parameters T = O (log(1 /π min )) = ˜ O ( d log(1 /r )) , M = π − min , τ = π min / (cid:107) π − δ F P ( T ) (cid:107) T V ≤ π min / . Combining this with (14), we obtain a distribution on discrete paths γ in Ω (with respect tothe adjacency relation ∼ ) such that each path has source F , E length( γ ) ≤ ˜ O ( m d /σ r ) · T, and the distribution of target( γ ) is within total variation distance π min / π . Letting G = { target( γ ) : length( γ ) ≤ E length( γ ) /φ } we immediately have that the diameter of G is at most˜ O ( m d /σ r ) · T /φ = ˜ O ( m d /σ rφ )and by Markov’s inequality π ( G ) ≥ − φ , as desired.12efore proceeding with the proofs of Lemmas 4.3 and 4.5, we collect the probabilisticnotation used throughout the sequel. We will often truncate on the following two highprobability events. Fix (cid:15) := m − d (15)and define: B := (cid:40) min S ∈ ( [ m ] d ) ,j ∈ [ m ] \ S dist( v j , aff( F S )) ≥ (cid:15) (cid:41) , C := (cid:26) max j ∈ [ m ] (cid:107) g j (cid:107) ≤ α (cid:27) . Note that whenever σ > m − d (which we may assume without loss of generality, as otherwisethe diameter is trivially at most 1 /σ ): P [ B ] ≥ − O ( m − d /σ ) ≥ − /m , (16)since the density of the component of v j orthogonal to aff( F S ) is bounded by 1 /σ and thereare at most m d facets. We also have P [ C ] ≥ − /m , (17)by standard Gaussian concentration and a union bound.We will repeatedly use that on C , we have the Hausdorff distance boundshdist( K, K ) ≤ α, hdist( K ( j ) , K ( j )0 ) ≤ α ∀ j ≤ m, (18)for α as in (12), since if x = (cid:80) j ≤ m c j ( a j + g j ) ∈ K for some convex coefficients c j then x = (cid:80) j ≤ m c j a j ∈ K and (cid:107) x − x (cid:107) ≤ α .For an index j ∈ [ m ] let ˆ g j := ( g , . . . , g j − , g j +1 , . . . g m ) and let K ( j ) = conv( v i : i (cid:54) = j ).Note that K ( j ) is a deterministic function of ˆ g j . Define the indicator random variables K S := { F S ∈ F d − ( K ) } , K ( j ) S := { F S ∈ F d − ( K ( j ) ) } for subsets S ∈ (cid:0) [ m ] d (cid:1) . It will be convenient to fix in advance a total order < on (cid:0) [ m ] d (cid:1) .We will occasionally refer to (cid:88) F ∈ F k ( K ) Vol k ( F )as the k − perimeter of K . π We will repeatedly use the following fact relating Hausdorff distance and containment ofconvex bodies. 13 emma 4.6 (Containment of Small Perturbations) . If hdist( K, K ) ≤ α for any two convexbodies and rB d ⊂ K , then (1 + 2 α/r ) − K ⊂ K ⊂ (1 + α/r ) K . Proof.
The second containment is immediate from K ⊂ K + αB d ⊂ K + ( α/r ) K . The condition hdist(
K, K ) ≤ α also implies K ⊂ K + αB d . To turn this into amultiplicative containment, we claim that ( r/ B d ⊂ K . If not, there is a point z ∈ ∂ ( r/ B d \ K . Choose a halfspace H supported at z containing K . Let y be a point in ∂ ( rB d ) at distanceat least r/ H and note that y ∈ K . But now dist( y, K ) ≥ dist( y, H ) ≥ r/ > α ,violating that K ⊂ K + αB d . Thus, we conclude that K ⊂ (1 + 2 α/r ) K , establishing thefirst containment. Proof of Lemma 4.3.
Condition on C . By ( S ) , ( R ), (18), and Lemma 4.6, we have K ⊃ (1 + 2 /d ) − K ⊃ ( r/ B d , (19)and also K ⊂ (1 + α ) B d . Consequently, the angle between any two adjacent facets F S , F T of K must satisfy | θ ST − π | = Ω(1 /r ) , which implies θ ST / ≤ tan( θ ST / ≤ O (1 /r ) θ ST (20)for all θ ST . Thus, for each facet S ∈ Ω: χ ( S ) / ≤ π ( S ) ≤ O ( r − ) χ ( S ) , (21)establishing Lemma 4.3(3).Equation (19) further implies: | ∂K || K | ≤ dr . By e.g. [Sch14, Section 4.2], we have the quermassintegral formulas: d · V ( K [ d − , B d [1]) = (cid:88) S ∈ Ω | F S | = | ∂K | , (22) (cid:18) d (cid:19) V ( K [ d − , B d [2]) = (cid:88) S S, T ∈ (cid:0) [ m ] d (cid:1) with S \ T = { j } : E (cid:20) B | F ST | csc θ ST K ST (cid:12)(cid:12)(cid:12)(cid:12) ˆ g j , C (cid:21) ≤ O ( d log m/σ ) · E (cid:20) | F ST | K ( j ) ST (cid:12)(cid:12)(cid:12)(cid:12) ˆ g j , C (cid:21) . (26) Proof. By trigonometry,csc θ ST = dist( v j , aff( F ST ))dist( v j , aff( F T )) ≤ v j , aff( F T )) , C , since K has diameter at most 2 + 2 α ≤ 3. The distance in the denominatorcan be rewritten asdist( v j , aff( F T )) = dist( g j + a j , aff( F T )) = dist( g j , aff( F T ) − a j ) = | h j − x T | where h j = (cid:104) g j , m T (cid:105) and x T = dist(0 , aff( F T ) − a j ) ≤ m T the unit normal to aff( F T ).Moreover, | F ST | K ST ≤ | F ST | K ( j ) ST with probability one conditional on ˆ g j since S ∩ T ∈ F d − ( K ) implies S ∩ T ∈ F d − ( K ( j ) ) as j / ∈ S ∩ T . Combining these facts, the left hand side of (26) is at most E (cid:20) B | F ST | K ( j ) ST v j , aff( F T )) (cid:12)(cid:12)(cid:12)(cid:12) ˆ g j , C (cid:21) = 3 | F ST | K ( j ) ST E (cid:20) B | h j − x T | (cid:12)(cid:12)(cid:12)(cid:12) ˆ g j , C (cid:21) , Notice that h j has density on R bounded by t (cid:55)→ √ πσ e − t / σ P [ (cid:107) g j (cid:107) ≤ α ] ≤ σ , and (cid:15) ≤ | h j − x T | ≤ | h j | + x T ≤ α < B , C , so the last conditionalexpectation is at most3 (cid:90) (cid:15) σt dt = 2(log(1 /(cid:15) ) + log 5) /σ ≤ O ( d log m/σ ) , completing the proof.The most technical part of the proof is the following ( d − d − K ( j ) is well-bounded by its ( d − Lemma 4.10 (Codimension 2 Perimeter versus Surface Area) . Assume ( R ) , ( S ) . For every j ∈ [ m ] : E C (cid:88) S 1] = 0 . The proof of Lemma 4.11 is deferred to Section 4.4.We rely on the following result of Dadush and Huiberts [DH20b, Theorem 1.13] (theyprove something a little stronger, but we use a simplified bound). Theorem 4.12 (Shadow Vertex Bound) . Suppose W is a fixed two dimensional plane and Q = conv { v , . . . , v m } where v i ∼ N ( a i , σ I ) with (cid:107) a i (cid:107) ≤ . Then E [ | F ( W ∩ Q ) | ] = O ( d . log ( m ) /σ ) . (28)Combining these two ingredients, we can prove Lemma 4.10. Proof of Lemma 4.10. Fix j ≤ m and recall that rB d ⊂ K ( j )0 ⊂ B d by ( R ). Conditioningon C , we also have hdist( K ( j ) , K ( j )0 ) ≤ α . Thus we may invoke Lemma 4.11 with L = K ( j )0 , L = K ( j ) , r = r , r = 1, and η = α = Ω( σ √ d log m ) to obtain a probability measure ν on two dimensional planes W ⊂ R d with the advertised properties; note that crucially W depends only on K and is independent of K . Let I (cid:15) be a maximal collection of disjoint( d − S (cid:15) of radius (cid:15) , with each S (cid:15) contained in some ( d − L .Notice that (cid:90) dν ( W ) E (cid:88) S The push-forward of ψ by Π x is absolutely continuous with respect to ψ (cid:48) withRadon-Nikodym derivative d ( ψ ◦ Π − x )( a (cid:48) ) dψ (cid:48) ( a (cid:48) ) = sin φ (cid:107) x − a (cid:107) d − , a (cid:48) = Π x ( a ) ∈ ∂B x where φ is the angle in [0 , π ] between the tangent plane to ∂ ˜ L at a and the line segment xa . Proof. An explicit Jacobian calculation given the definition of Π x and smoothness of ∂ ˜ L givesthe result. 20 emma 4.14. Let z (cid:54)∈ aff( S (cid:15) ) be a point such that Π z is injective on S (cid:15) . Let V be a randomtwo-dimensional subspace. Then P ( V + z ∩ S (cid:54) = 0) = µ (Π z ( S )) /A d − where µ is the Hausdorff measure of Π z (aff( S )) and A d − = µ (Π z (aff( S ))) (half the surfacearea of S d − ).Proof. Since aff( S (cid:15) ) misses z , we have that aff( { z } ∪ S (cid:15) ) is d − z is smooth and injective on aff( S (cid:15) ) so Π z ( S (cid:15) ) itself is d − V + z ) (cid:54)⊂ aff( { z } ∪ S ) , which occurs with probability 1. Then ( V + z ) ∩ aff( { z } ∪ S ) is aline through z . By symmetry, the intersection of that line with B d + z will be a uniformlyrandom antipodal pair. Exactly one point from each pair will fall in Π z (aff( S )). Thus, theevent we care about is the event that y ∈ Π z ( S ) where y is sampled uniformly from µ .The following Lemma takes a and a (cid:48) to be fixed, and depends only on the randomness of V . Lemma 4.15. Let a be a point not in aff( S (cid:15) ) and a (cid:48) = Π x ( a ) . Let θ be the angle between S (cid:15) and the ray emanating from a through x . Then, for a uniformly random − plane V , P ( V + a ∩ S (cid:15) (cid:54) = ∅ ) = Vol d − ( S (cid:15) ) A d − cos θ (cid:107) x − a (cid:107) d − (1 + O ( d(cid:15) )) and P ( V + a (cid:48) ∩ S (cid:15) (cid:54) = ∅ ) = Vol d − ( S (cid:15) ) A d − (cos θ )(1 + O ( d(cid:15) )) . where the convergence is uniform in a, a (cid:48) . In particular, the ratio of the above two quantitiesis (cid:107) x − a (cid:107) d − .Proof. We apply Lemma 4.14 twice, both times with S (cid:15) playing the role of S . The first timewe take a to play the role of z , and the second time a (cid:48) . This gives P ( V + a ∩ S (cid:15) (cid:54) = ∅ ) = µ a (Π a ( S (cid:15) )) /A d − and P ( V + a (cid:48) ∩ S (cid:15) (cid:54) = ∅ ) = µ a (cid:48) (Π a (cid:48) ( S (cid:15) )) /A d − µ a , µ a (cid:48) are the Hausdorff measures on Π a (aff( S (cid:15) )) , Π a (cid:48) (aff( S (cid:15) )) respectively. Let µ (cid:48) bethe surface measure on aff( S (cid:15) ). Then the Radon-Nikodym derivatives of µ (cid:48) and the pull-backsof µ a and µ a (cid:48) are d ( µ a ◦ Π a )( y ) dµ (cid:48) ( y ) = cos θ ay (cid:107) y − a (cid:107) d − and d ( µ a (cid:48) ◦ Π a (cid:48) )( y ) dµ (cid:48) ( y ) = cos θ a (cid:48) y (cid:107) y − a (cid:48) (cid:107) d − where θ ay , θ a (cid:48) y are the angles between S (cid:15) and the rays from a, a (cid:48) to y respectively. This allowsus to compute µ a (Π a ( S (cid:15) )) = ( µ a ◦ Π a )( S (cid:15) ) = (cid:90) S (cid:15) cos θ ay (cid:107) y − a (cid:107) d − dµ ( y ) = Vol d − ( S (cid:15) ) cos θ ax (cid:107) x − a (cid:107) d − (1 + O ( d(cid:15) )) . The same is true for a (cid:48) in place of a . Note that θ a (cid:48) x = θ ax = θ , and that (cid:107) x − a (cid:48) (cid:107) = 1. Thatgives the desired result. Lemma 4.16 (Reduction to ∂B x ) . Let W be as above and let W (cid:48) be a uniformly randomtwo dimensional plane through a uniformly random point a (cid:48) chosen from ∂B x . Then forsufficiently small (cid:15) > (depending only on L ): Vol d − ( ∂ ˜ L ) P [ W ∩ S (cid:15) (cid:54) = ∅ ] ≥ r r η Vol d − ( ∂B x ) P [ W (cid:48) ∩ S (cid:15) (cid:54) = ∅ ] Proof. Note that a, a (cid:48) miss aff( S (cid:15) ) with probability 1, so we implicitly condition on that eventin the following.Vol d − ( ∂ ˜ L ) P [ W ∩ S (cid:15) (cid:54) = ∅ ]= (cid:90) P [ W ∩ S (cid:15) (cid:54) = ∅ (cid:12)(cid:12) a ] dψ ( a )= (cid:90) P [ W ∩ S (cid:15) (cid:54) = ∅ (cid:12)(cid:12) a (cid:48) = T ( a )] d ( ψ ◦ Π − x )( a (cid:48) ) by invertibility of Π x (4.13)= (cid:90) P [ W ∩ S (cid:15) (cid:54) = ∅ (cid:12)(cid:12) a (cid:48) ] sin φ (cid:107) x − a (cid:107) d − dψ (cid:48) ( a (cid:48) ) by Claim 4.13 ≥ (cid:90) (cid:0) P [ W (cid:48) ∩ S (cid:15) (cid:54) = ∅ (cid:12)(cid:12) a (cid:48) ](1 / (cid:1) sin φ (cid:107) x − a (cid:107) dψ (cid:48) ( a (cid:48) ) by Claim 4.15, for sufficiently small (cid:15) ≥ r r η (cid:90) P [ W (cid:48) ∩ S (cid:15) (cid:54) = ∅ (cid:12)(cid:12) a (cid:48) ] dψ (cid:48) ( a (cid:48) ) = r r η Vol d − ( ∂B x ) P [ W (cid:48) ∩ S (cid:15) (cid:54) = ∅ ] , where in the final inequality we have used (cid:107) x − a (cid:107) ≥ η − η = η and sin φ ≥ r r because˜ L ⊃ L ⊃ ( r − η ) B d ⊃ ( r / B d and ˜ L ⊂ ( r + η ) B d ⊂ r B d . Lemma 4.17 (Intersection Probability for ∂B x ) . P ( W (cid:48) ∩ S (cid:15) (cid:54) = ∅ ) = Vol d − ( S (cid:15) ) A d − C d √ d (1 + O ( d(cid:15) )) for some constant C d = Θ(1) depending on d . roof. Using iterated expectation, we can write P ( W (cid:48) ∩ S (cid:15) (cid:54) = ∅ ) = E ( P ( W (cid:48) ∩ S (cid:15) (cid:54) = ∅ : a (cid:48) ))where the outer expectation is over the randomness of a (cid:48) and inner probability over V . Theinner probability is given by 4.15 asVol d − ( S (cid:15) ) A d − cos( θ a (cid:48) x )(1 + O ( d(cid:15) )) . The only dependence on a (cid:48) is in cos( θ a (cid:48) x ). However, by symmetry of the distribution of a (cid:48) , θ a (cid:48) x might as well measure the angle between a uniform random vector selected from ∂B x andany fixed line. Thus E [cos θ a (cid:48) x ] = C d / √ d, for some constant C d = Θ(1) depending on d .We can now complete the proof of Lemma 4.11. Combining Lemmas 4.16 and 4.17, wehave for sufficiently small (cid:15) > d − ( ∂ ˜ L ) P [ W ∩ S (cid:15) ] ≥ r r η Vol d − ( ∂B x ) · Ω(1) √ dA d − Vol d − ( S (cid:15) ) = Ω (cid:18) r d / r η (cid:19) Vol d − ( S (cid:15) )since Vol d − ( ∂B x ) /A d − = 2 π/d , as desired. In this section we explain how any instance of the smoothed unit LP model may be reducedto one for which ( S ) , ( R ) hold with parameter (11), incurring only a polynomial loss in m . Proof of Theorem 1.2. The idea is to add the noise vector g j as the sum of two independentGaussians g j, ∼ N (0 , σ ) and g j, ∼ N (0 , σ ) with σ guaranteeing roundedness and σ supplying the necessary anticoncentration and concentration for the main part of the proof.Given σ < /d , set σ = m σ and σ + σ = σ and let K be equal to K perturbed by g only. Applying Lemma 4.18 toeach K ( j )0 and taking a union bound, we have K ( j )1 ⊃ rB d ∀ j ≤ m, r = Ω( σm − ) = Ω( σ m ) , with probability 1 − O ( m − ). Since σ < /d , another union bound reveals that K ⊂ B d − O ( m − ); let K = K / 2. Now K is an instance of the smoothed unitLP model, ( K , σ ) satisfy ( R ) with r = Ω( σ m ) = Ω( σ/m ), and6 (cid:112) d log mσ = o ( r/m ) , so ( K , σ ) also satisfy ( S ), establishing (11) with the role of ( K , σ ) now played by ( K , σ ).Invoking Theorem 4.1, we conclude that with probability 1 − /m , for every φ ∈ (0 , G ⊂ Ω with π ( G ) ≥ (1 − φ ) π (Ω) and facet diameter˜ O ( m d / ( σ/m ) ( σ/m ) φ ) = poly( m, d ) /σ φ. Moreover, by Lemma 4.3(3), we have χ ( G ) ≤ π ( G ) ≤ φ · π (Ω) ≤ φ · O ( m /σ ) χ (Ω) , so we conclude that χ ( G ) ≥ (1 − ψ ) χ (Ω) for ψ = O ( m φ/σ ). Rewriting the diameterbound in terms of ψ yields the desired conclusion. The probability may be upgraded to1 − / poly( m ) by Remark 4.2 Lemma 4.18 (Roundedness of Smoothed Polytopes) . Suppose we have m ≥ d + 1 points a , . . . , a m ∈ R d , and these are perturbed to v , . . . , v m by adding independent g j ∼ N (0 , σ I d ) to each respective a j . Then, with probability at least − O ( m − ) , the convex hull K of v , . . . , v m contains a ball of radius r in ≥ Ω( σ m − ) . Proof. Without loss of generality, taking the first d + 1 points a i , we may assume that m = d + 1. Then K is the convex hull of d + 1 points v , . . . , v d +1 . The probability that theaffine span of these points equals R d is 1. Let r in be the inradius of K ; by Lemma 4.7, wehave r in ≥ min i dist( v i , aff( F i )) d + 1 . Let us now fix an i and obtain and obtain a probabilistic lower bound on dist( v i , aff( F i )) d +1 . Reorderthe points (if necessary) so that i = d + 1. It now follows that given the the affine span A of the points v , . . . , v d and given a d +1 , the distribution of dist( v d +1 , A ) is the same asthe distribution of | ˜ g + dist( a d +1 , A ) | , where ˜ g ∼ N (0 , σ ) has the distribution of a onedimensional Gaussian with variance σ . However, the probability that | ˜ g + dist( a d +1 , A ) | isless than σ m − is at most O ( m − ). Therefore, by the union bound, P (cid:104) min i dist( v i , aff( F i )) > σd − (cid:105) > − O (cid:0) m − (cid:1) . It follows that P (cid:20) r in > σm − d + 1 (cid:21) > − O (cid:0) m − (cid:1) , as desired. 24 cknowledgments We thank Daniel Dadush, Bo’az Klartag, and Ramon van Handel for helpful commentsand suggestions on an earlier version of this manuscript. We thank Ramon van Handelfor pointing out the important reference [Izm10]. We thank the IUSSTF virtual center on“Polynomials as an Algorithmic Paradigm” for supporting this collaboration. References [Ale37] Alexander Alexandroff. 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Pick M to be minimal with respect to c in the sense that for each i there exists a point x with (cid:104) m i , x (cid:105) = b i but (cid:104) m j , x (cid:105) < b j for j (cid:54) = i ; note that this implies F c ( c ) > i ≤ N . Let R ( c ) := diag( c / | F ( c ) | , · · · , c N / | F N ( c ) | )and ˜ H ( c ) = R / ( c ) H ( K ( c )) R / ( c ). By Sylvester’s inertial law, it suffices to prove thestatement for ˜ H instead of H .If K ( c ) is simple, then the proof of the statement can be found in [CEKMS19, Proposition3]. In fact, it is shown there that for simple polytopes the positive eigenvalue of ˜ H ( c ) isequal to d − 1. If K ( c ) is not simple, let u be a vector with independent entries sampleduniformly from [0 , δ > K ( c + δu ) is simple withprobability 1. Then we will show that the entries of ˜ H ( c ) are continuous in a neighborhoodof c so that for each (cid:15) , picking δ small enough guarantees | ˜ H ( c ) − ˜ H ( c + δu ) | ≤ (cid:15) entry-wise.If we have both these claims, then we will have a sequence of matrices of the form˜ H ( c ) + (cid:16) ˜ H ( c + δu ) − ˜ H ( c ) (cid:17) which approaches ˜ H ( c ) with each matrix in the sequence having exactly one positive eigenvalue,which is d − 1. Since the spectrum is a continuous function of the matrix, we conclude ˜ H ( c )has exactly one positive eigenvalue, which is d − K ( c + δu ) is simple : For any collection S ⊂ [ n ] if indices, let M S (resp. c S , u S ) be thesubmatrix of M (resp. c , u ) with row i included if and only if i ∈ S . Fix any S with | S | > d .27hen rk( A S ) ≤ d , so the column space of A S has measure 0 in R | S | , meaning it misses c S + u S with probability 1.˜ H ( c ) is continuous : It suffices to argue that H ( K ( c )) and R ( c ) are each continuous. Firstwe argue that K ( c ) has N facets everywhere in a neighborhood of c . Since M is is minimalfor c , the facets of K ( c ) correspond exactly to the rows of M . Perturbations of c do notchange the rows of M , so this correspondence is preserved if M is minimal for c + δu . Tothis end, let x ( i ) be such that (cid:104) m i , x ( i ) (cid:105) = c i and (cid:104) m j , x ( i ) (cid:105) < c j for j (cid:54) = i . Then simply pick δ < 12 min i min j (cid:54) = i (cid:0) c j − (cid:104) m j , x ( i ) (cid:105) (cid:1) max i,j |(cid:104) m j , m i (cid:105)| / (cid:107) m i (cid:107) and let ˜ x ( i ) = x ( i ) + u i δ m i (cid:107) m i (cid:107) . Then (cid:104) m i , ˜ x ( i ) (cid:105) = c i + u i δ but (cid:104) m j , ˜ x ( i ) (cid:105) = (cid:104) m j , x ( i ) (cid:105) + u i δ (cid:104) m j , m i (cid:105)(cid:107) m i (cid:107) < (cid:104) m j , x ( i ) (cid:105) + 12 ( c j − (cid:104) m j , x ( i ) (cid:105) ) < c j < c j + u j δ as desired. This also rules out discontinuities in | F i ( c ) | , so R ( c ) is indeed continuous.Now for H ( K ( c )). For pairs i (cid:54) = j such that m i is a multiple of m j , we have F ij = 0independently of c . Otherwise, note that csc( θ ij ) , cot( θ ij ) are finite and depend only on M ,so it suffices to show that F ij is continuous in c . To this end, F ij itself is the volume of a d − { x (cid:48) ∈ R d − : M (cid:48) x (cid:48) ≤ f ( c ) } where M (cid:48) ∈ R ( N − × ( d − and f : R N → R N − is an affine function of c . Specifically, take i = 1 and j = 2 without loss of generality and let Q be the product of two Householder transformationssuch that M Q = αβ γv v M (cid:48) for some scalars α, β, γ , v , v ∈ R N − , and M (cid:48) ∈ R ( N − × ( d − . Let Q T x = ( x (cid:48) , x (cid:48) , x (cid:48) ) T where x (cid:48) , x (cid:48) are scalars and x (cid:48) contains the remaining d − Q T x . Let c = ( c , c , c ) T where c , c are scalars and c contains the remaining N − c . Then K ( c ) = { x ∈ R d : M x ≤ c } = { x : M QQ T x ≤ c } = x : αβ γv v M (cid:48) x (cid:48) x (cid:48) x (cid:48) ≤ c c c . Restricting to the points where the first two constraints tight means setting x (cid:48) = c /α and x (cid:48) = ( c − βx ) /γ . We therefore have F = Vol( { x (cid:48) | M (cid:48) x (cid:48) ≤ c − x (cid:48) v − x (cid:48) v } )which is continuous in cc