{"id":"d790105c-2e92-49f5-a657-8806a0215e94","arxiv_id":"2607.03815","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A simplex-based symmetry ratio recovers Minkowski measure after affine invariance, improves its stability, characterizes simplices by outer additivity, and sharply bounds the ratio for low-depth polytopes.","lead":"The paper defines a simplex-based ratio measuring how a convex body sits inside a simplex versus its opposite, then proves this recovers the classical Minkowski symmetry measure after taking affine images. It improves stability estimates, characterizes simplices by an additivity property, and shows low-depth polytopes built by sums and convex hulls stay far from simplices, limiting certain neural-network constructions.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript's central geometric claim for neural-network depth is self-contained and elementary once the support-function formulae of Section 2 are granted. The only non-elementary external ingredient (Akopyan–Karasev) is confined to the stability half of the paper and does not underwrite the sharp depth bound or the inapproximability of h_Delta. The reader's identification of Lemma 18 as the weakest technical step is accurate for the stability theorems, but that step is not load-bearing for the strongest claim the reader himself highlights. Consequently the ACCEPT verdict with high confidence stands; no adjustment is required.","tokens_in":19401,"tokens_out":435,"duration_ms":3538,"concrete_test":"Independently recompute rho_Delta on the explicit depth-d construction given at the end of the proof of Theorem 7 (the (2^d-1)-dimensional simplex formed by 2^d vertices of Delta_n) and confirm equality holds; simultaneously verify that the induction step of Theorem 7 never calls Lemma 18 or Theorem 20.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption flag on Lemma 18 (smooth/strictly-convex density + inductive covering + Akopyan–Karasev) is real but non-load-bearing for the strongest claim (Theorem 7). Theorem 7 is proved by induction on depth using only Theorems 9–10 (support-function averaging for Minkowski sums and the elementary max-bound for conv of two sets) together with the base cases for points and zonotopes; those arguments never invoke d_Delta, the upper bound of Lemma 18, or the Kadets-type theorem. The same holds for the ensuing inapproximability of h_Delta. Lemma 18 is used only for the improved Minkowski-stability statement (Theorem 5), which is independent of the depth result. No internal gap appears in the inductive-covering verification (Claim 21) or the density reduction that would propagate into Theorem 7.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The paper defines a simplex-based (non-affine-invariant) measure of symmetry ρ_Δ(L) := λ_{-Δ}(L)/λ_Δ(L) via smallest containing homothets of the regular simplex and its negative. It proves four main results: (1) the classical Minkowski measure m*(K) equals the supremum of ρ_Δ over all origin-barycentered simplices in the affine hull (Theorem 2); (2) an improved stability estimate: m*(L) ≥ n-ε implies Banach–Mazur distance at most 1/(1-ε) to a simplex (Theorem 5), via the comparison n/d_Δ(L) ≤ ρ_Δ(L) ≤ n-1 + 1/d_Δ(L) (Theorem 3); (3) a characterization that a convex body K is outer-additive (λ_K additive under Minkowski sum) if and only if K is a simplex (Theorem 6); (4) for the depth complexity of polytopes generated by Minkowski sums and convex hulls of unions, the sharp bound ρ_Δ(P) ≤ 2^d - 1 whenever depth(P) = d (Theorem 7), with the consequence that low-depth polytopes cannot approximate simplices and that m*(P) is likewise bounded (Theorem 8), plus an inapproximability transfer to support functions of input-convex ReLU networks.","tokens_in":19633,"tokens_out":1098,"duration_ms":14321,"significance":"The work cleanly unifies classical convex geometry (Minkowski measure, stability, indecomposability/characterizations of simplices) with a modern computational model of polytopes arising from ReLU/ICNN expressivity. The depth bound is sharp, elementary (induction via support-function averaging for sums and a max-bound for conv-union), and yields a uniform geometric obstruction to approximating the simplex (hence the max function) by low-depth constructions; this is a genuine contribution beyond asymptotic indecomposability results of Shephard type. The outer-additivity characterization is novel and self-contained. The stability improvement is asymptotically sharper than Schneider’s earlier bound and holds for the full range ε < 1. All arguments are written in full detail from support functions and Carathéodory; the only external input is the Akopyan–Karasev Kadets-type theorem, used solely for the upper half of Theorem 3 (hence only for stability, not for the depth results).","major_comments":[],"minor_comments":[{"comment":"In the proof of Lemma 18 the reduction to smooth strictly convex bodies is asserted by density; a one-sentence justification that both ρ_Δ and d_Δ are continuous in the Hausdorff metric (or that the inequality passes to the limit) would make the argument fully self-contained.","section":"§4.2, Lemma 18"},{"comment":"Claim 21 verifies inductiveness of the covering by a general J-covering argument; the base case and the two inductive steps are clear, but a short remark that the covering may have fewer than n+1 sets (k ¤ n+1) does not affect the application of Theorem 20 would help the reader.","section":"§4.2, Claim 21"},{"comment":"The sharpness construction for Theorem 7 (a (2^d-1)-dimensional simplex built from vertices of Δ) is only sketched; spelling out the inductive construction of the vertex set would make the equality case completely explicit.","section":"§6.3, proof of Theorem 7"},{"comment":"Notation for the model simplex alternates between Δ, Δ^n and Δ_n; a single consistent choice (e.g., Δ_n throughout) would improve readability.","section":"throughout"},{"comment":"In the introduction the phrase “measures of symmetry capture how close … and are (assumed to be) affine invariant” is slightly awkward; the parenthetical can be dropped or rephrased.","section":"§1.1"}],"recommendation":"accept","confidential_remarks":"The depth-complexity application is the most novel part and fits well with recent STOC/ICLR work on ICNN depth; the pure convex-geometry results (outer additivity, improved stability) are solid but more incremental. No citation or novelty concerns. The manuscript is already in good shape for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is four solid results from one elementary ratio ρ_Δ = λ_{-Δ}/λ_Δ. The strongest and cleanest is the depth bound: any polytope of depth d under Minkowski sum and conv-of-union satisfies ρ_Δ ≤ 2^d - 1 (sharp), so simplices cannot be approximated by shallow polytopes. That argument is pure induction on the two operations via support-function averaging (Theorems 9–10) and never touches the Kadets-type theorem. The same bound immediately gives the Minkowski-measure version and the inapproximability statement for h_Δ that the ICNN people care about.\n\nWhat is new: the ratio itself, its identification with m* after affine maximization (Theorem 2), the dimension-free stability d_BM ≤ 1/(1-ε) when m* ≥ n-ε (Theorem 5), and the if-and-only-if characterization that outer-additive bodies are exactly the simplices (Theorem 6). The outer-additivity proof is geometric and self-contained: a local parallelogram obstruction plus induction on dimension. The stability improvement is real and asymptotically better than Schneider’s earlier bound.\n\nSoft spots are minor and correctly localized. Lemma 18 (the upper comparison of ρ_Δ with d_Δ) does rely on density to smooth strictly convex bodies plus the Akopyan–Karasev inductive-covering theorem. That is the only non-elementary external input, and it is used only for the stability statement; it does not load-bear the depth results. The inductive-covering verification (Claim 21) looks correct on a careful read. Citations are appropriate and light on self-reference.\n\nThis is for people who work on measures of symmetry or on geometric lower bounds for ReLU/ICNN depth. The math is fully written out, the definitions are non-circular, and the strongest claim stands on elementary ground. I would send it to a serious referee without hesitation.","headline":"Clean convex-geometry paper: new simplex ratio, sharp outer-additivity characterization, improved stability, and a sharp depth obstruction for approximating simplices that is independent of the Kadets input.","tokens_in":20218,"tokens_out":565,"would_cite":true,"duration_ms":5220,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","52A40","52B11"],"pacs":[],"model":"grok-4.5","headline":"A simplex ratio recovers the Minkowski measure of symmetry, characterises simplices by outer additivity, and proves low-depth polytopes cannot approximate them.","keywords":["measure of symmetry","Minkowski measure","simplex","Banach-Mazur distance","outer additivity","depth complexity","input convex neural networks","polytopes"],"falsifier":"Produce either a non-simplex that is outer additive, or a depth-d polytope in dimension n with d<ceil(log_{2}(n+1)) whose simplex ratio exceeds 2^d-1, or a body with Minkowski measure at least n-ε whose Banach–Mazur distance to the nearest simplex is strictly larger than 1/(1-ε).","tokens_in":20318,"feed_emoji":"△","tokens_out":1140,"duration_ms":19512,"temperature":0.7,"pith_summary":"The paper introduces a simple ratio that compares the smallest positive and negative simplex homothets needed to cover a convex body. Taking the supremum of this ratio over all centered simplices recovers the classical Minkowski measure of symmetry. With that identification the authors improve the known stability theorem: any body whose Minkowski measure is within ε of the maximum n is at most 1/(1-ε) away from a simplex in Banach–Mazur distance. They also prove that the only convex bodies for which the outer-containment size is additive under Minkowski summation are the simplices themselves. Finally, for polytopes assembled by nested Minkowski sums and convex hulls of unions—the operations that define depth for input-convex ReLU networks—they obtain the sharp bound that the simplex ratio is at most 2^d-1 at depth d, so simplices cannot be approximated by shallow constructions.","feed_headline":"Low-depth polytopes cannot approximate simplices","feed_subtitle":"A simple ratio shows they stay at least a fixed distance away, with a sharp 2^d-1 bound","key_machinery":"The ratio ρ_Δ of the two outer coefficients with respect to a fixed simplex Δ, expressed via the average of support-function values on the n+1 facet normals of a centered simplex. This single number is simultaneously an affine-invariant proxy for Minkowski symmetry, an additive invariant that characterises simplices, and a depth-monotone quantity for polytopes.","core_discovery":"The simplex-based measure ρ_Δ(L)=λ_-Δ(L)/λ_Δ(L) has an affine-invariant envelope equal to the Minkowski measure of symmetry. The same quantity is controlled by the two operations that generate depth-d polytopes (Minkowski sum never increases it beyond the worst summand; convex hull of a union at most adds the two values plus one), yielding the sharp upper bound ρ_Δ(P)≤2^d-1 whenever the depth of P is d. Consequently any polytope of depth less than log_{2}(n+1) stays a definite distance from the simplex.","pith_inferences":["The same logarithmic-depth lower bound is likely to obstruct shallow approximation of any continuous piecewise-linear function that is complete for the max function, limiting the expressivity of shallow input-convex networks more broadly.","Outer additivity supplies a new axiomatic route to recognising simplices that may be useful when studying indecomposability or approximation by Minkowski sums.","The inductive-covering technique used for the upper bound on ρ_Δ can probably be reused to obtain quantitative stability statements for other classical measures of asymmetry."],"forward_implications":["Any convex body whose Minkowski measure is at least n-ε lies at Banach–Mazur distance at most 1/(1-ε) from a simplex.","A convex body makes outer containment size additive under Minkowski sum if and only if it is a simplex.","Every polytope of depth complexity d satisfies ρ_Δ≤2^d-1, and the bound is attained by certain simplices of dimension 2^d-1.","Support functions of depth-d polytopes remain a definite L1 distance away from the max-of-(n+1)-linears function whenever d is smaller than log_{2}(n+1).","The same depth obstruction applies to every body of full Minkowski measure n, not only to the regular simplex."],"fun_headline_variants":["Depth-d polytopes stay at least 2^d-1 from any simplex","Simplex measure sharply limits low-depth polytope approximation","Only simplices are outer-additive under sum and convex hull","ρ_Δ forces depth log(n) before polytopes near a simplex","Minkowski symmetry envelope equals affine hull of simplex ratio"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The upper comparison between the simplex ratio and the inner-to-outer homothety ratio rests on reducing to a smooth strictly convex body and then applying a Kadets-type theorem for inductive coverings of space by supporting half-spaces.","fun_headline_variants_meta":{"raw":{"variants":["Depth-d polytopes stay at least 2^d-1 from any simplex","Simplex measure sharply limits low-depth polytope approximation","Only simplices are outer-additive under sum and convex hull","ρ_Δ forces depth log(n) before polytopes near a simplex","Minkowski symmetry envelope equals affine hull of simplex ratio"]},"model":"grok-4.5","effort":"low","cost_usd":0.005294,"raw_usage":{"total_tokens":1561,"prompt_tokens":922,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":52940000,"prompt_tokens_details":{"text_tokens":922,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":545,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":922,"tokens_out":94,"duration_ms":5021,"temperature":1.0,"reasoning_tokens":545,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T23:45:25.365037+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce either a non-simplex that is outer additive, or a depth-d polytope in dimension n with d<ceil(log_{2}(n+1)) whose simplex ratio exceeds 2^d-1, or a body with Minkowski measure at least n-ε whose Banach–Mazur distance to the nearest simplex is strictly larger than 1/(1-ε).","supporting_citations":[],"review_version":1}