Equality in the Fradelizi–Paouris–Schütt second-moment inequality
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Original abstract
Fradelizi, Paouris, and Schütt proved a sharp lower bound for the second moment of a convex body whose extreme points lie outside an open Euclidean ball. They characterized equality for polytopes and conjectured the same characterization for arbitrary convex bodies. We prove this conjecture: equality holds precisely for nondegenerate simplices whose vertices lie on the bounding sphere. The proof extends their simplex argument by an almost-everywhere partition into countably many simplices with extreme-point vertices. An exact nonnegative deficit identity retains the equality information for this partition.
The idea in plain English
The second moment measures the average squared distance of a body from a fixed origin. Fradelizi, Paouris and Schütt proved a sharp lower bound when every extreme point stays on or outside a specified sphere. They also found exactly which polytopes attain equality.
This note extends that equality characterization to arbitrary convex bodies in every dimension n≥1. Equality holds precisely for nondegenerate simplices whose vertices lie on the sphere. The simplex need not be regular, its centroid need not be the origin, and it need not contain the origin.
The proof keeps a finite or countable collection of simplices whose vertices are actual extreme points of the original body. An exact sum of nonnegative deficits preserves the information needed to recognize equality. A second argument retains a fixed pair of simplices to show that a body which is not a simplex has a strictly positive deficit.
The inequality, simplex moments and polytope equality mechanism are established FPS results. The contribution presented here is the extension retaining equality for general bodies; a bounded literature comparison does not certify first priority.
The precise result
Let n≥1 and r>0, and let K⊂ℝⁿ be compact and convex with nonempty interior. Assume ‖v‖₂≥r for every actual extreme point v∈ext K. Write |K| for n-dimensional Lebesgue volume, g_K=|K|⁻¹∫_K x dx for its centroid, and C₂(K)=|K|⁻¹∫_K ‖x‖₂² dx for its normalized second moment.
The full inequality is C₂(K)≥[r²+(n+1)‖g_K‖₂²]/(n+2). Equality holds if and only if K is a nondegenerate n-simplex and all its vertices have norm r. The radius hypothesis is non-strict and anchored at the specified origin. If every extreme-point norm is strictly greater than r, equality is impossible.
The weaker inequality C₂(K)≥r²/(n+2) has equality exactly for the same inscribed simplices with centroid zero. In dimension one the equality interval is [−r,r]. Regularity is not required for the strengthened inequality: the triangle with vertices (1,0),(0,1),(−1,0) has centroid (0,1/3), C₂=1/3 and equality at r=1.
If an initial extreme-vertex simplex S differs from K, retain the next visible-facet simplex T formed from an outside extreme point u. With a=|S|>0, b=|T|>0 and v_i the opposite vertex, the deficit has the positive lower bound ab‖u−v_i‖₂²/[(n+1)(n+2)|K|(a+b)]. This is a finite body-dependent strictness witness.
Why the result holds
Classical finite-dimensional extreme-point representation and separability give a countable dense choice of actual extreme points. Insert them successively into finite hulls. Strictly visible facet shells split into simplices using only their existing vertices; retaining all earlier cells produces disjoint interiors and covers the body up to its null boundary. No closedness of the extreme set or compatible face subdivisions is required.
For each cell S_j, put w_j=|S_j|/|K|, c_j its centroid and q_j the mean squared norm of its vertices. FPS’s established simplex moment formula and weighted variance give the exact identity Δ_r(K)=[Σ_j w_j(q_j−r²)]/(n+2)+[(n+1)/(n+2)]Σ_j w_j‖c_j−g_K‖₂². Compactness makes all the moment series absolutely convergent, and every displayed term is nonnegative.
If the deficit vanishes, every positive-volume cell has all vertices on the sphere and centroid g_K. Two nondegenerate simplices with that same centroid would share an interior point, contradicting their disjoint interiors. Hence the body is one closed simplex. Conversely, the simplex moment formula proves equality for every inscribed nondegenerate simplex.
The separate finite route retains the same two positive-volume simplices S and T through every larger finite hull. Completing the square in their centroid terms gives the fixed positive witness before taking limits. Bounded moment convergence then preserves it; strictness of unrelated approximating polytopes alone would not suffice.
Scope and limitations
The theorem uses the Euclidean norm about the specified origin, ordinary Lebesgue volume, compactness and nonempty n-dimensional interior. Degenerate or unbounded bodies, other norms and other probability weights are outside its stated scope.
No centroid-zero, origin-containment, regular-simplex, smooth-boundary, finite-extreme-set or exposed-extreme-point assumption is made. Only the weaker bound without the centroid term requires centroid zero for equality. The countable construction is an analytic existence proof, not a finite algorithm for arbitrary bodies.
The retained finite witness depends on the particular body and chosen simplices. It is not a uniform quantitative stability modulus in a shape distance. Finite exact integration checks corroborate formulas and examples rather than proving the universal theorem.
FPS already proved the inequality, simplex moments, finite vertex decomposition and common-centroid rigidity for polytopes. The original FPS and Oberwolfach formulations identify the historical question; no claim about the current status or wording of an online catalogue is needed. Bounded bibliographic coverage does not certify absence of later equivalent work, first disclosure or current worldwide openness.
Status and AI use
AI tools were used extensively in research, proof development, drafting and verification. The manuscript was prepared with OpenAI Codex from a candidate supplied by the author from earlier AI-assisted work, with separate AI verification agents. Internal AI-assisted checks are not external human or journal peer review. This unrefereed preprint has no formal proof-assistant certification; the named author is responsible for its claims and citations.
The manuscript and Zenodo’s declared publication date are 23 September 2026, version 1.0.0. Zenodo created the record on 26 September 2026 UTC. These edition dates do not establish historical priority.
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Substantive collaboration on independent verification, a meaningful convex-geometric extension, or development toward journal submission is welcome. Authorship and responsibilities would reflect actual research and writing contributions and mutual agreement.
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Cite this paper
Alec Kriebel. (2026). Equality in the Fradelizi–Paouris–Schütt second-moment inequality (Version 1.0.0, 23 September 2026) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22983138
Background and prior work
- Fradelizi, Paouris and Schütt, Simplices in the Euclidean ball: The established centroid-corrected inequality, simplex moment formulas, finite vertex partition and equality rigidity for polytopes, followed by the general-body equality conjecture. These results and their proof mechanism are prior work.
- Fradelizi, joint work with Paouris and Schütt, Oberwolfach report contribution: Printed pages 2907–2909 record the same historical inequality and general-body equality question. A historical conjecture statement does not establish current worldwide open status.
- Bertsekas, Convex Optimization Theory: Classical finite-dimensional extreme-point representation. The note also gives the elementary shell and barycentric arguments explicitly.