A Spectral Approach to Polytope Diameter

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a"giant component"of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schr\"odinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.


Introduction
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 widely open, with the best known upper bounds being (m − d) log 2 d−log 2 log d+O(1) ( [23], see also [15,25]) and O(m) for fixed d ( [16,3]); the best known lower bound is (1 + ϵ)m for some ϵ > 0 when d is sufficiently large [19]. Given this situation, there has been interest 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 ( [4,7,12,8], [6,22,27,9]) 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 separately over the past century, with several celebrated achievements including the Alexandrov-Fenchel inequality [1] and more generally the Hodge-Riemann relations for certain algebras associated with simple polytopes [24]. One consequence of this theory is that a certain Schrödinger operator (weighted adjacency matrix plus diagonal) associated with the graph of every bounded polytope has a spectral gap [14] (see Definition 5 and Theorem 6). We use this fact to make progress on Q1 and Q2. In the first setting, we show the following theorem.
▶ Theorem 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 has diameter O(d 2 ∆ 2 log(m∆)).
Theorem 1 follows from a more geometric result (Theorem 10) stated in terms of the angles between the facets of the polar of P , which is proven in Section 3. It compares favorably (in the regime m ≪ 2 d ) with the best previously known result of this kind due to [8], who achieved a bound of O(d 3 ∆ 2 ) (though notably their ∆ depends only on subdeterminants of A and not on b). However, our diameter bound is nonconstructive whereas [8] show how to efficiently find a path between any two vertices of P ; we refer the reader to the introduction of that paper for a more thorough discussion of previous work in this vein (originally initiated by [11,4]). At a high level, the reason we are able to save a factor of d in comparison with previous works is that they rely on combinatorial expansion arguments, whereas we use spectral expansion, which is amenable to a "square root" improvement using Chebyshev polynomials.
Regarding Q2, the study of diameters of random polytopes began with the influential work of Borgwardt [5,6], who considered A with i.i.d. standard Gaussian entries and b = 1. Borgwardt showed the following "for each" guarantee: for any fixed objective functions c, c ′ ∈ R d , the combinatorial distance between the vertices x, x ′ of P maximizing ⟨c, x, ⟩, ⟨c ′ , x ′ ⟩ is at most O(d 3/2 √ log m) in expectation, provided m → ∞ sufficiently rapidly. This type of result was extended to the "smoothed unit LP" model by Spielman and Teng in the seminal work [22]; in this model one takes where v j ∼ N (a j , σ 2 ) for some fixed vectors a 1 , . . . , a m normalized to have ∥a j ∥ ≤ 1. The original poly(m, d, σ −1 ) path length bound of [22] was improved and simplified in [10,27,9]; 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 that all of these results provide "for each" guarantees: at best they bound the distance between a single pair of vertices, not between all pairs.
Our second contribution is to prove that for the smoothed unit LP model, most pairs of vertices in P are polynomially (in m, d, σ −1 ) close with high probability, where most is defined with respect to a certain locally defined measure on the vertices known as the mean curvature measure χ 2 in convex geometry (see [21,20]; we recall the definition in Section 4). In the language of random graph theory, this means that the graph of P likely contains a "giant component" with respect to χ 2 which is of small diameter.
▶ Theorem 2. Assume P is a random polytope sampled from the smoothed LP model. Let χ 2 denote the mean curvature measure on the facets of P • , which corresponds naturally to a measure on the set of vertices of P , denoted Ω. Then with probability at least 1 − 1/poly(m), 108:3 We prove Theorem 2 in Section 4.5, where we deduce it from a more refined theorem (Theorem 13, which includes explicit powers of m, d) for a certain class of well-rounded polytopes. The idea of the proof is to consider a certain continuous time Markov chain whose states are the vertices of P . This chain automatically has a large spectral gap by Theorem 6 and the main challenge is to bound its average transition rate. This is carried out in Sections 4.2-4.4 and involves further use of the Alexandrov-Fenchel inequalities, tools from integral geometry, Gaussian anticoncentration, and an application of the shadow vertex bound of [9].
▶ Remark 3. It was pointed out to us by one of the anonymous referees and by D. Dadush that there is a "folklore" result that the average distance between a random pair of vertices (chosen by optimizing two uniformly random objective functions) of P as above is polynomial in m, d, σ −1 ; this is seen by a Fubini type argument and the shadow vertex bounds of [22,10,27,9]. Our result is incomparable since it considers a different measure on the vertices.
▶ Remark 4 (Expansion of Polytopes). There has been a sustained interest in studying the expansion of graphs of combinatorial polytopes beginning with [17] which conjectured that all 0/1 polytopes have expanding graphs. The recent breakthrough [2] resolved this conjecture for the special case of matroid polytopes using techniques related to high dimensional expanders and the geometry of polynomials, which may be described as capturing "discrete log-concavity". The present work, in contrast, uses "continuous log-concavity" (stemming from the Brunn-Minkowski inequality) to control the spectral gaps of certain matrices associated with the graphs of polytopes with favorable geometric properties. We note that the Hirsch conjecture is already known to hold for 0/1 polytopes [18].

Preliminaries and Notation
We recall some basic terminology and facts regarding polytopes; the reader may consult [21,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 >0 be a bounded polytope containing the origin in its interior. Its polar is the polytope where a T 1 , . . . , a T m 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 otherwise noted, "facet" refers to a codimension-1 facet. The polar of a simple polytope is simplicial and vice versa.
The 1−dimensional facets of a polytope are called edges, and are all line segments when it is bounded. The vertex diameter of a bounded polytope P is the diameter of the graph of its vertices and edges. Two (d − 1)-facets of a polytope are adjacent if their intersection is a (d − 2)-facet of the polytope. The facet diameter of a polytope K is the diameter of the graph 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 • .

108:4 A Spectral Approach to Polytope Diameter
We use dist(·, ·) to denote the Euclidean distance between two subsets of R d , and to denote the Hausdorff distance between two sets. We to denote the mixed volume of j copies of K and d − j copies of L for convex bodies K, L ⊂ R d . The Alexandrov-Fenchel inequalities imply that these are log-concave, in the sense that for j 1 We use C to denote absolute constants whose value may change from line to line, unless specified otherwise.

Eigenvalues of the Hessian and Spectral Gaps
In this section, we recall that a certain matrix associated with every bounded polytope has exactly one positive eigenvalue.
where F ij is the intersection of facets i and j, and θ ij is the angle between the vectors normal to 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} with respect to the slack vector c > 0. (see [21,Chapter 4]). Log-concavity of the volume implies that this Hessian has exactly one positive eigenvalue. Izmestiev [14] has shown via an approximation argument that this remains true for the formal Hessian of any polytope. ▶ Theorem 6 (Theorem 2.4 of [14]). H(K) has exactly one positive eigenvalue for any bounded polytope K.
We include a self-contained proof of Theorem 6 in the Appendix of the arxiv version of this paper for completeness 1 .
We will apply Theorem 6 to certain matrices derived from the formal Hessian and the following diagonal scaling, which plays an important role in the remainder of the paper. ▶ Definition 7. Let K, F ij , θ ij be as in Definition 5. Then let D(K) denote the N × N positive diagonal matrix with entries (D(K)) ii = k F ik tan(θ ik /2). Note that θ ik ̸ = π whenever F ik = 0 since parallel facets of a convex polytope cannot intersect.  The left corresponding to this eigenvalue is D1.

Proof.
Observe that H is "nearly" a graph Laplacian in the sense that: where we have used the identity 1−cos θ sin θ = tan(θ/2). By Sylvester's inertia law, the signature of H matches that of which must therefore have exactly one positive eigenvalue by Theorem 6. However, L ⪰ 0 and L1 = 0, so by Sylvester's law −D −1/2 LD −1/2 ⪯ 0 with at least one eigenvalue equal to zero. Thus, D −1/2 HD −1/2 has exactly one eigenvalue equal to one, with eigenvector D 1/2 1 and the rest of the eigenvalues nonpositive, establishing the first claim. The second claim follows from (7) and the similarity of D −1 L and D −1/2 LD −1/2 . ◀

Diameter in Terms of Angles and Bit Length
In this section we use the spectral gap bound of Lemma 8(1) to give a bound on the diameter of a polytope specified by integer constraints. We begin by generalizing the argument of [26], who used Chebyshev polynomials to control the diameter of regular and biregular graphs in terms of their spectra, to handle the matrix D −1/2 HD −1/2 by appropriately controlling its negative entries and top eigenvector.
Proof. Note that if M ∈ span(I, A, · · · , A pk ) then e T i M e j ̸ = 0 if and only if there is a path in G from i to j of length at most pk. To this end, consider T p k (A) where T k is the degree k Chebyshev Polynomial of the first kind. If we find that T p k (A) ̸ = 0 entry-wise, then we can conclude the diameter of G is at most pk. Let be the spectral decomposition of A. Let | · | denote the entry-wise absolute value. Then We would therefore have T p k (A) ̸ = 0 entry-wise if N is smaller then the smallest absolute entry of vv T T p k (1 + g), which is lower bounded by v 2 min T p k (1 + g) ≥ v 2 min (1 + 0.9 gk 2 ) p . It suffices to pick k = 1.1/ √ g and p = log 2 (N/v 2 min ).

The angle between any two adjacent facets is contained in
Proof. Every co-dimension 2 facet can be written as the convex hull of some subset of size at least d − 1 of the vertices a 1 /b 1 , · · · , a m /b m . Without loss of generality, say that is the smallest co-dimension 2 facet. Then its volume is: and we have used that the determinant of a nonsingular integer matrix is at least one. On the other hand, P o is contained inside the ℓ 2 ball of radius d 1/2 2 B ≤ 2 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 ≤ 2 dB+d log d , establishing (1).
Regarding the angles, consider without loss of generality a facet F = conv(a 1 /b 1 , . . . , a d /b d ) and a vertex a j , j > d of a facet adjacent to F . Observe that the angle θ between the normals to these adjacent facets satisfies: .
The numerator is at most the distance between two vertices of P • , which is at most The denominator is given by By the adjugate formula, the entries of M −1 are of magnitude at most ∆ d−1 , so we have Combining these bounds yields To obtain (2), observe that that Finally, we can prove the bound advertised in the introduction.
Proof of Theorem 1. Applying Theorem 10, Lemma 12 (3), and the relation ∆ 1 ≤ 2 B , we find that the diameter of P is at most

Smoothed Analysis
In this section we consider the "smoothed unit LP" model defined in (1). Suppose P 0 is a fixed polytope specified as I T C S 2 0 2 2 108:8 A Spectral Approach to Polytope Diameter for some vectors ∥a j ∥ ≤ 1, and consider the random polytope Denote the polars of P 0 and P by Note that K is simplicial with probability one, so each of its k-dimensional facets has exactly k + 1 vertices. We will use the notation F S := conv{v j : j ∈ S} to denote facets k+1 : F S is a k-dimensional facet of K to denote the set of all facets of K. The k-dimensional volume of a facet F S , S ∈ F k (K) will be denoted by |F S | or Vol k (F S ). We will often abbreviate F S∩T as F ST for adjacent S, T . For two S, T ∈ [m] d , let θ ST ∈ (0, π) denote the angle between the unit normals u S , u T to F S , F T , respectively; note that almost surely θ ST ̸ = 0, π for every S, T ∈ [m] d . We will pay special attention to the set of (d − 1)-facets of K, which we denote as Define the measures χ 2 , π, δ : Ω → R ≥0 by It will be convenient to make two further technical assumptions on K 0 and σ for the proofs of our results; in Section 4.5 we will show that any instance of the smoothed unit LP model may be reduced to one satisfying both assumptions with parameter incurring only a poly(m) loss in the diameter. Let K (j) 0 = conv(a i : i ̸ = j) be the polytope obtained from K 0 by deleting vertex a j . (R) Roundedness of Subpolytopes: There is an r ∈ (0, 1) such that for every j ≤ m: (S) Smallness of σ: The main result of this section is the following "almost-diameter" bound with respect to the measure π.
where L is as in (5). The corresponding Markov semigroup has stationary distribution proportional to D1 = π(·) by Lemma 8 (2); call the normalized stationary distribution π(·) := π(·)/π(Ω) The first property is that the stationary distribution π is (in a quite mild sense) nondegenerate, with high probability. Apart from being essential in our proofs, this relates the measure π to well-studied measures in convex geometry such as the surface measure and mean curvature measure χ 2 (·), clarifying the meaning of Theorem 13. The proof of Lemma 15 appears in Section 4.1.
▶ Lemma 15 (Non-degeneracy of π). Assume (S), (R). With probability at least 1 − 1/m 2 : The second property is that Q (almost surely) has a spectral gap of at least one, by Lemma 8(2). This implies that the chain (13) mixes rapidly to π (in the sense of continuous time) from any well-behaved starting distribution. In particular let us say that a probability measure p on Ω is an M -warm start if Let ℓ 2 (π) denote the inner product space on defined on R Ω , where the inner product is given by ⟨f, g⟩ ℓ2(π) := S∈Ω π(S)f (S)g(S), and let ℓ 1 (π) be the corresponding ℓ 1 space. Let Π be the Ω × Ω diagonal matrix whose S th diagonal entry is π(S). We define the density of p with respect to π to be the the vector with entries p(S) π(S) . We omit the proof of the following standard fact.

108:10 A Spectral Approach to Polytope Diameter
The third and final property is a bound on the rate at which the continuous chain makes discrete transitions between states. Let J avg denote the average number of state transitions made by the continuous time chain in unit time, from stationarity, and note that as the diagonal entries of the generator Q are equal to −δ(S)/π(S). The most technical part of the proof is the following probabilistic bound.
By Lemma 15(1), the distribution δ F0 concentrated on F 0 is π −1 min −warm with probability 1 − 1/m 2 . Invoking Lemma 16 with starting distribution δ F0 and parameters T = O(log(1/π min )) =Õ(d 2 log(1/r)), M = π −1 min , τ = π min /2 we have Combining this with (14), we obtain a distribution on discrete paths γ in Ω (with respect to the adjacency relation ∼) such that each path has source F 0 , and the distribution of target(γ) is within total variation distance π min /2 of π. Letting G = {target(γ) : length(γ) ≤ 2Elength(γ)/ϕ} we immediately have that the diameter of G is at most and by Markov's inequality π(G) ≥ 1 − ϕ, as desired. ◀ Before proceeding with the proofs of Lemmas 15 and 17, we collect the probabilistic notation used throughout the sequel. We will often truncate on the following two high probability events. Fix ϵ := m −5d (15) and define: Note that whenever σ > m −d (which we may assume without loss of generality, as otherwise the diameter is trivially at most 1/σ): since the density of the component of v j orthogonal to aff(F S ) is bounded by 1/σ and there are at most m d facets. We also have by standard Gaussian concentration and a union bound.
We will repeatedly use that on C, we have the Hausdorff distance bounds for α as in (12), since if x = j≤m c j (a j + g j ) ∈ K for some convex coefficients c j then x 0 = j≤m c j a j ∈ K 0 and ∥x − x 0 ∥ ≤ α.
For an index j ∈ [m] letĝ j := (g 1 , . . . , g j−1 , g j+1 , . . . g m ) and let K (j) = conv(v i : i ̸ = j). Note that K (j) is a deterministic function ofĝ j . Define the indicator random variables d . It will be convenient to fix in advance a total order < on [m] d . We will occasionally refer to Vol k (F ) as the k−perimeter of K.

Nondegeneracy of π
We will repeatedly use the following fact relating Hausdorff distance and containment of convex bodies.

108:12 A Spectral Approach to Polytope Diameter
Proof. The second containment is immediate from To turn this into a multiplicative containment, we claim that (r/2)B d 2 ⊂ K. If not, there is a point z ∈ ∂(r/2)B d 2 \ K. Choose a halfspace H supported at z containing K. Let y be a point in ∂(rB d 2 ) at distance at least r/2 from H and note that y ∈ K 0 . But now dist(y, K) ≥ dist(y, H) ≥ r/2 > α, violating that K 0 ⊂ K + αB d 2 . Thus, we conclude that K 0 ⊂ (1 + 2α/r)K, establishing the first containment. ◀ Proof of Lemma 15. Condition on C. By (S), (R), (18), and Lemma 18, we have and also K ⊂ (1 + α)B d 2 . Consequently, the angle between any two adjacent facets F S , F T of K must satisfy (20) for all θ ST . Thus, for each facet S ∈ Ω: By the Alexandrov-Fenchel inequality with β = 1/2: By Alexandrov-Fenchel with β = 1/(d − 1), we also have The last step follows from the fact that (1). Combining these inequalities with (20), (21), we conclude that: establishing Lemma 15 (2). The event B implies that for every S ∈ Ω:

Average Jump Rate Bound
In this section we establish the following Lemma, which immediately implies Lemma 17 by Proof.
Proof. By trigonometry, conditional on C, since K has diameter at most 2 + 2α ≤ 3. The distance in the denominator can be rewritten as where h j = ⟨g j , m T ⟩ and x T = dist(0, aff(F T ) − a j ) ≤ 4 for m T the unit normal to aff(F T ). Moreover, Combining these facts, the left hand side of (26) is at most Notice that h j has density on R bounded by and ϵ ≤ |h j − x T | ≤ |h j | + x T ≤ 4 + α < 5 conditioned on B, C, so the last conditional expectation is at most completing the proof. ◀ The most technical part of the proof is the following (d − 2)-perimeter estimate, whose proof is deferred to Section 4.3. The conceptual meaning of this estimate is that on average, the (d − 2)-dimensional surface area of a random facet of K (j) is well-bounded by its (d − 1)-dimensional volume.

Proof of Lemma 22
The key step in the proof is to show that for any well-rounded polytope L 0 , there a distribution on two-dimensional planes W such that the (d − 2)-perimeter of every nearby polytope L is accurately reflected in the average number of vertices of W ∩ L. Since this number of vertices is small in expectation by [9], we can then conclude that the codimension 2 perimeter is small.
In this section and the next only, the variable ϵ will refer to a quantity tending to zero (as opposed to the definition (15)).

▶ Lemma 23 (Quadrature by Planes). Let
There there is a probability distribution on two dimensional planes W in R d such that for sufficiently small ϵ > 0 the following holds uniformly over every polytope L with at most m d facets satisfying

108:16 A Spectral Approach to Polytope Diameter
The proof of Lemma 23 is deferred to Section 4.4. We rely on the following result of Dadush and Huiberts [9, Theorem 1.13] (they prove something a little stronger, but we use a simplified bound).
▶ Theorem 24 (Shadow Vertex Bound). Suppose W is a fixed two dimensional plane and Combining these two ingredients, we can prove Lemma 22.
Proof of Lemma 22. Fix j ≤ m and recall that rB . Conditioning on C, we also have hdist(K (j) , K (j) 0 ) ≤ α. Thus we may invoke Lemma 23 with L 0 = K (j) 0 , L = K (j) , r 1 = r, r 2 = 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 0 and is independent of K. Let I ϵ be a maximal collection of disjoint by the "Moreover" part of Lemma 23 |FST |K The integrand in the first expression above above is at most since each set in [m] d−2 appears as the intersection of at most m 2 adjacent pairs S, T . Therefore by Theorem 24 the first expression above is bounded above by O(m 2 d 5/2 log 2 (m)/σ 2 ). Rearranging yields implying the desired conclusion. ◀

Proof of Lemma 23
We provide an explicit construction for the distribution of W . LetL = L 0 + 2ηB d 2 and note that its boundary ∂L is smooth; let ψ be the d − 1-dimensional surface measure on ∂L.
This equals both the d − 1 dimensional Hausdorff measure and the Minkowski content of ∂L. Then let W = V + a where a is a point sampled according to ψ, and V is sampled by taking the span of two Gaussian vectors (or any radially symmetric random vectors). In order to compute P(V + a ∩ S ϵ ̸ = ∅), it will help to first reduce it to the related probability where a ′ is sampled uniformly from the unit ball which shares a center with S ϵ . In particular, let x be the center of S ϵ and denote B x = B d 2 + x. Let ψ ′ be the d − 1-dimensional Hausdorff measure on ∂B x . Then we will reduce to the case of P(V + a ′ ∩ S ϵ ̸ = ∅) for a ′ sampled according to ψ ′ . For any z, define the radial projection Π z by Note that Π x is a bijection between ∂L and ∂B x since every ray originating from x intersects ∂L in exactly one point because x is in the interior ofL, which is convex.
▷ Claim 25. The push-forward of ψ by Π x is absolutely continuous with respect to ψ ′ with Radon-Nikodym derivative 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 gives the result. ◁ ▶ Lemma 26. Let z ̸ ∈ aff(S ϵ ) be a point such that Π z is injective on S ϵ . Let V be a random two-dimensional subspace. Then where µ is the Hausdorff measure of Π z (aff(S)) and A d−2 = µ(Π z (aff(S))) (half the surface area of S d−2 ).
Proof. Since aff(S ϵ ) misses z, we have that aff({z} ∪ S ϵ ) is d − 1 dimensional. On the other hand, Π z is smooth and injective on aff(S ϵ ) so Π z (S ϵ ) itself is d − 2 dimensional. Condition on (V + z) ̸ ⊂ aff({z} ∪ S) , which occurs with probability 1. Then (V + z) ∩ aff({z} ∪ S) is a line through z. By symmetry, the intersection of that line with B d 2 + z will be a uniformly random antipodal pair. Exactly one point from each pair will fall in Π z (aff(S)). Thus, the event we care about is the event that y ∈ Π z (S) where y is sampled uniformly from µ. ◀ I T C S 2 0 2 2

108:18 A Spectral Approach to Polytope Diameter
The following Lemma takes a and a ′ to be fixed, and depends only on the randomness of V . ▶ Lemma 27. Let a be a point not in aff(S ϵ ) and a ′ = Π x (a). Let θ be the angle between S ϵ and the ray emanating from a through x. Then, for a uniformly random 2−plane V , where the convergence is uniform in a, a ′ . In particular, the ratio of the above two quantities is ∥x − a∥ d−2 .
Proof. We apply Lemma 26 twice, both times with S ϵ playing the role of S. The first time we take a to play the role of z, and the second time a ′ . This gives where µ a , µ a ′ are the Hausdorff measures on Π a (aff(S ϵ )), Π a ′ (aff(S ϵ )) respectively. Let µ ′ be the surface measure on aff(S ϵ ). Then the Radon-Nikodym derivatives of µ ′ and the pull-backs of µ a and µ a ′ are where θ a y , θ a ′ y are the angles between S ϵ and the rays from a, a ′ to y respectively. This allows us to compute The same is true for a ′ in place of a. Note that θ a ′ x = θ a x = θ, and that ∥x − a ′ ∥ = 1. That gives the desired result. ◀ ▶ Lemma 28 (Reduction to ∂B x ). Let W be as above and let W ′ be a uniformly random two dimensional plane through a uniformly random point a ′ chosen from ∂B x . Then for sufficiently small ϵ > 0 (depending only on L 0 ): Proof. Note that a, a ′ miss aff(S ϵ ) with probability 1, so we implicitly condition on that event in the following.
for some constant C d = Θ(1) depending on d.
Proof. Using iterated expectation, we can write where the outer expectation is over the randomness of a ′ and inner probability over V . The inner probability is given by 27 as The only dependence on a ′ is in cos(θ a ′ x ). However, by symmetry of the distribution of a ′ , θ a ′ x might as well measure the angle between a uniform random vector selected from ∂B x and any fixed line. Thus for some constant C d = Θ(1) depending on d. ◀ We can now complete the proof of Lemma 23. Combining Lemmas 28 and 29, we have for sufficiently small ϵ > 0:

Removing Assumptions (S),(R)
In this section we explain how any instance of the smoothed unit LP model may be reduced to one for which (S), (R) hold with parameter (11), incurring only a polynomial loss in m.
Proof of Theorem 2. The idea is to add the noise vector g j as the sum of two independent Gaussians g j,1 ∼ N (0, σ 2 1 ) and g j,2 ∼ N (0, σ 2 2 ) with σ 1 guaranteeing roundedness and σ 2 supplying the necessary anticoncentration and concentration for the main part of the proof. Given σ < 1/d, set σ 1 = m 8 σ 2 and σ 2 1 + σ 2 2 = σ 2 and let K 1 be equal to K 0 perturbed by g 1 only. Applying Lemma 30 to each K so (K 2 , σ 2 ) also satisfy (S), establishing (11) with the role of (K 0 , σ) now played by (K 2 , σ 2 ). Invoking Theorem 13, we conclude that with probability 1 − 1/m 2 , for every ϕ ∈ (0, 1) there is a subset G ⊂ Ω with π(G) ≥ (1 − ϕ)π(Ω) and facet diameter 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 1 , . . . , v d+1 . The probability that the affine span of these points equals R d is 1. Let r in be the inradius of K; by Lemma 19, we have Let us now fix an i and obtain and obtain a probabilistic lower bound on dist(vi,aff(Fi))

d+1
. Reorder the points (if necessary) so that i = d + 1. It now follows that given the the affine span A of the points v 1 , . . . , v d and given a d+1 , the distribution of dist(v d+1 , A) is the same as the distribution of |g + dist(a d+1 , A)|, whereg ∼ N (0, σ 2 1 ) has the distribution of a one dimensional Gaussian with variance σ 2 1 . However, the probability that |g + dist(a d+1 , A)| is less than σ 1 m −4 is at most O m −4 . Therefore, by the union bound, It follows that as desired. ◀