Papers

A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, \kappa\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}

For a triangle of area \(T\), a planar corner-area characterization states that an interior point \(M\) lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through \(M\) parallel to the sides have areas \(T_1,T_2,T_3\) satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If \(S\) is a nondegenerate \(n\)-simplex of volume \(V\) and \(V_1(M),\ldots,V_{n+1}(M)\) are the volumes of the facet-parallel corner simplices determined by \(M\), then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where \(E_J(S)\) is the John ellipsoid of \(S\). We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.

For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic \[ \mathcal C_n(\alpha)-1\asymp_n(1-\alpha)^{2/(n+1)} \qquad(\alpha\uparrow1) \] for the uncentered Hardy–Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak $(1,1)$ growth rate $(k-1)^{-(n-1)/2}$ as $k\downarrow1$, uniformly over Radon measures. We also obtain the corresponding $L^p$ and Fefferman–Stein bounds.

A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the Gömböc of Domokos and Várkonyi), and Lángi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.

We study Sobolev and $BV$-extension properties of planar subsets. In particular, we prove that fat Sierpiński carpets that support a weak $(1,1)$-Poincaré inequality are $W^{1,1}$-extension sets, and we provide an explicit linear extension operator for them. We also show that for planar domains satisfying a weak $(1,1)$-Poincaré inequality the $W^{1,1}$-extension property and the $BV$-extension property are equivalent. Finally, as a consequence of the previous result, we show that if a simply-connected planar domain is Ahlfors regular and supports a weak $(1,1)$-Poincaré inequality, then it is a (linear) $W^{1,1}$-extension domain.

For $0\le m\le n$, let $c_{m,n}$ be the infimum of scales at which the Vietoris–Rips filtration of the round sphere $\mathbb{S}^m$ admits a continuous odd map from $\mathbb{S}^n$. A quantitative Borsuk–Ulam theorem gives $c_{m,n}/2\le d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$, and it was asked whether equality always holds. Using a synchronized product-measure lift of the spherical join, we construct continuous odd maps between Vietoris–Rips metric thickenings with target scale equal to the maximum input scale. Iteration gives $c_{m+d,n+d}\le c_{m,n}$ for every integer $d\ge0$. More generally, the finite join law adds $m_i+1$ and $n_i+1$ separately and bounds the resulting $c$-value by $\max_i c_{m_i,n_i}$. For $0\le r<\pi$, the pairs $(k,\ell)$ with $c_{k-1,\ell-1}\le r$ are closed under addition, so Fekete's lemma gives a limit for the maximal admissible $\ell/k$ as $k\to\infty$. This structure, exact values, and projective-code estimates give finite and asymptotic bounds. If $1\le m_j<n_j$, $m_j\to\infty$, and $\log(n_j+1)=o(m_j)$, then $c_{m_j,n_j}\to\pi/2$. At fixed $r\in(\pi/2,\pi)$, projective-code and projective-covering estimates give lower and upper exponential bounds for the largest $n$ with $c_{m,n}\le r$. Together with exact values, these yield a sharp transition: the largest $n$ admitting an odd map is $m$ at scale $\pi/2$ and grows exponentially with $m$ at each fixed $r\in(\pi/2,\pi)$. We finally return to the conjectural equality. We introduce a Gromov–Hausdorff-stable invariant from chromatic numbers of Borsuk-graph filtrations. Combining projective-code upper bounds for $c_{m,n}$ with chromatic lower bounds from spherical-cap volumes, we show that if $m_j<n_j$, $m_j\to\infty$, and $n_j/m_j\to\lambda\ge9$, then $d_{\mathrm{GH}}(\mathbb{S}^{m_j},\mathbb{S}^{n_j})>c_{m_j,n_j}/2$ for all sufficiently large $j$, producing infinitely many counterexamples.

We identify the rotational symmetry centers in the spectre tiling. We find that both 2-fold and 3-fold symmetry centers are coexisting, with two different types of 3-fold centers. Local and global symmetry operations are distinguished, and characterized in terms of substitution systems. We introduce a decomposition of the spectre tiling in terms of three metatiles named diabolo, 2-arms and 3-arms, valid at any substitution level. We propose geometrical constructions for generating patches of the spectre tiling with 2-fold and 3-fold global rotational symmetry.

We prove that the covering dimension of a metric space with curvature bounded above equals its geometric dimension, without a separability assumption. This resolves a conjecture formulated by Kleiner that goes back to Gromov's work.

We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a Hölder function and a function that suitably “undoes” the Hölder regularity. We show simple ways to construct such “de-Höldering” factorizations. If the identity mapping on a metric space $\mathcal{M}$ admits a de-Höldering factorization through a metric space $\mathcal{Z}$ that has a conical geodesic bicombing, then the class of metric spaces from which one can extend $\mathcal{Z}$-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for $\mathcal{M}$-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every $L_1$-valued Lipschitz function from a subset of $\ell_2$ can be extended to a Lipschitz function that takes values in $L_1$ and is defined on all of $\ell_2$, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size $n$ and quality $O(\sqrt{\log n})$, improving Moitra's 2009 bound. We also show that for every metric space $\mathcal{Z}$ that has a conical geodesic bicombing, any metric transform of a metric space $\mathcal{M}$ has $\mathcal{Z}$-valued Lipschitz extension modulus at most a universal constant multiple of the $\mathcal{Z}$-valued Lipschitz extension modulus of $\mathcal{M}$ itself, improving the 2002 bound of Brudnyi and Shvartsman.

In this work we present the following general principle: every positive homogeneous functional on an abstract convex cone satisfying a suitable Brunn-Minkowski inequality gives rise to an information functional -defined as the quotient of the functional by its relative surface area- that satisfies a linear Brunn-Minkowski inequality. We also establish the dual counterpart of this principle for functionals satisfying an analogous convexity property, yielding reverse linear Brunn-Minkowski inequalities for the associated information functionals. Furthermore, we show that homogeneity is necessary within this principle by providing a counterexample to the expected result for the information functional associated to the standard Gaussian measure. As applications, we recover, via a different, first-order argument, a classical result by Giannopoulos, Hartzoulaki and Paouris (2002) for the ratio of two consecutive quermassintegrals, together with a characterization of the equality case. The same argument extends to the setting of mixed discriminants of symmetric positive definite matrices. Finally, our approach also yields a new $L_p$ extension ($p \geq 1$) of the Brunn-Minkowski inequality by Dembo, Cover and Thomas for the classical information functional, where the relative surface area is expressed in terms of Lutwak's first $L_p$ quermassintegral.

Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteq\Delta_n^d$. For $d\geq 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\geq 5$ variables and for polynomials of degree $d\geq 4$ in $n\geq 4$ variables. We identify the two minimal obstructions, at $(n,d)=(4,4)$ and $(n,d)=(5,3)$, and propagate them throughout the failure region by degree and variable aggregation.

We study to what extent the majorisation order between a probability measure and its $1$-Lipschitz image is preserved when both measures undergo Gaussian convolution. We show that the majorisation order is fully preserved in dimensions $n\leq 2$, and obtain dimension-dependent partial preservation in higher dimensions. The perspective taken is that majorisation between densities amounts to comparison through internal energies satisfying a certain pressure condition. Accordingly we introduce a notion of iterated nonnegativity of pressure, closely related to iterated pressures arising in optimal transport, placing the majorisation order within a graded hierarchy of internal energy comparisons. Using the observation that the volume of Euclidean neighbourhoods can be detected from internal energy measurements along the heat flow, we show that our results imply several principal known cases of the Kneser–Poulsen conjecture.

We study the infinite-dimensional Gaussian pyramid and its quotients by the global sign flip and the $U(1)$-Hopf action. We resolve affirmatively a long-standing problem posed by Tomohiro Fukaya around 2014: these three limiting geometries are pairwise non-similar, meaning that no positive rescaling makes any two of them coincide.

We prove the future-completeness conjecture of Gödeke and Rendall for expanding spatially homogeneous vacuum spacetimes in four spatial dimensions. More generally, an expansion-energy estimate gives future timelike and null geodesic completeness for homogeneous Einstein equations coupled to maps into complete Riemannian targets, through nine spatial dimensions for vanishing potential. In arbitrary dimension we obtain an affine-length criterion which includes positive potential floors and potentials that decay sufficiently slowly along the scalar trajectory. For future-global homogeneous solutions, the critical nine-dimensional bound also holds for a general stress tensor in an explicit normal energy-condition window, including a broad range of perfect fluids. At the opposite time end, we establish a quotient-stable $C^0$-inextendibility criterion for two-step nilpotent cosmologies. Its proof combines boundary localization with an optimal, intrinsically defined allocation of the central correction in timelike homotopies, and continues to apply when the diameter of compact spatial slices collapses. Exact Heisenberg solutions realize both conclusions. On every fixed compact Bianchi~II quotient, the expanding invariant vacuum data split into an open dense set with globally $C^0$-inextendible developments and a codimension-two locally rotationally symmetric locus with analytic compact Cauchy horizons. We determine the chronology-violating region of the exceptional extensions and quantify the exponentially small proper-time scale on which transverse anisotropy replaces the horizon by a singular end.

We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our approach does not rely on differentiating RNP-targets, and uses instead decomposability bundles and a careful blow-up analysis. We also present decomposability bundles in a way which avoids the mention of Alberti representations and generalizes the approach of Alberti–Marchese to measures in RNP-spaces. We moreover study fragment-wise differentiability into RNP-targets, give a new ${\rm Lip}-{\rm lip}$-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces $(X,\mu)\subset\ell^2$ whose Gromov–Hausdorff tangents are Hausdorff limits of $r^{-1} (X-x)$ in $\ell^2$ as $r\to 0$.

A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound $\mu = \frac{2}{3} + \frac{2}{3\sqrt{3}}$. This resolves negatively a question posed by K. Bezdek and Z. Lángi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.

In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron $T_1$ and an isosceles, nonregular tetrahedron $T_2$ whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In $T_1$ and in $T_2$ there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, $T_1$ and $T_2$ are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron $T_2$ has $AB=CD=1$ and $AC=AD=BC=BD=\sqrt{(21+4\sqrt6)/45}$, while $T_1$ is given by a root of a sextic with Galois group $S_6$ and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than $95^\circ$; the tetrahedron $T_2$ has two dihedral angles of about $98.7^\circ$. For isosceles Morley tetrahedra we conjecture that $(AB^2-CD^2)(AC^2-BD^2)(AD^2-BC^2)\ge0$ and that $AB=CD$ forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with $AC=BD$ whose Morley tetrahedron satisfies $A'B'=B'C'=C'D'=D'A'$ has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.

We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, the main ideas and details of the construction, and its relationship with existing techniques.

For every $n\geqslant1$, the equilateral dimension of the Heisenberg group $\fH^n=\C^n\times\R$ with the Korányi metric is $2n+2$.

We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.

We study bounded ratios and optimal bounding constants among the normalized coefficients of ternary Lorentzian polynomials. For every fixed $M$-convex support and in arbitrary degree, we give an explicit presentation of the bounded-ratio cone in terms of quadratic Hessian slices. We then express the optimal bounding constants through a variational formula combining local support functions with linear compatibility constraints between slices. For full support, we determine all compatibility relations in arbitrary degree; in degree three, this yields explicit optimal constants for every two-generator section. Finally, we compare the resulting Lorentzian bounds with those for volume polynomials and rank-three matroid basis profiles.

Let $P_d(132,213)$ be the convex hull of the permutations in $S_d$ that avoid $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial, defined by $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$. We give an explicit lattice equivalence between this polytope, a path-Laplacian deficit polytope, and twice a translated chain-poset permutahedron. The resulting face description and triangulation give a self-contained proof of Davis and Sagan's cubicality and volume conjecture, with normalized volume $2^{d-1}d^{d-3}$ for $d\ge2$. For $d\ge3$, the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences. The refined Eulerian expansion of $H_d(t)$ has a negative coordinate already at $d=7$. We construct a compatible cone of adjacent signed differences that accommodates this obstruction. Its largest uniform repair parameter is $2-\sqrt{3}$, and membership reduces to explicit geometric-tail inequalities. An exact dataset and reconstruction algorithm verify these inequalities through $d=1000$. A marked-component identity, discrete smoothing, and Darroch's mode theorem (Ann. Math. Statist. 35 (1964), 1317–1321, Theorem 4) prove them for every $d\ge2^{72}$. Thus $H_d(t)$ has only negative real zeros for $3\le d\le1000$ and for $d\ge2^{72}$, while $H_1(t)=H_2(t)=1$. Uniform real-rootedness in the intervening range remains open.

We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.

We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel $\psi:[-1,1]\to\R$, let $T_\psi$ be the associated zonal convolution operator on the sphere and let $\cB_\psi K$ be the corresponding surface-area observable of a convex body $K\subset\R^d$. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the $L^\infty$ norm of the potential discrepancy $T_\psi(\mu-\sigma)$, for every normalized signed directional measure $\mu$. Hence universal exactness is equivalent to $\mu-\sigma\in\ker T_\psi$. Funk–Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of $T_\psi$ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For $\psi_p(s)=|s|^p$ we obtain a complete classification: even powers give precisely weighted real projective designs, whereas non-even powers admit no finite signed atomic rule that is exact for every convex body. For the classical Cauchy kernel $|s|$, exactness fails but spherical $t$-designs give a uniform $O(t^{-1})$ relative surface-area error. We also derive exact projection-moment identities for rectifiable submanifolds.

Let $S_{26}$ denote the supremum of the sum of the radii of 26 pairwise interior-disjoint closed disks of positive radius contained in the unit square. We prove $2.635983084917607783\ldots \le S_{26}\le \sqrt{\frac{2577}{2500}+\frac{11}{\sqrt3}}=2.716919756099889566\ldots$. The lower bound is accompanied by an exact rational feasibility certificate. For the upper bound, we assign the disks their clipped power cells, convert cell area into a reciprocal exterior-angle cost, and combine this local inequality with Euler side counts. The extremal case for the resulting relaxation has four boundary transitions and a four-cell boundary cycle; its continuous angle allocation is reduced to a concave one-variable problem. All finite numerical inequalities are rigorously enclosed by a reproducible certificate. The argument does not determine $S_{26}$ exactly, but gives a nontrivial global upper bound for a benchmark whose best-known packing is asymmetric.

It is known that any convex cone in a general vector space is the positive cone of a preorder compatible with the vector space, and that any convex set is an affine slice of a convex cone. We therefore treat convex sets order-theoretically, as affine slices of positive cones. We find that faces are sets of strong minima, and we generalize functionals to oriented maps valued in arbitrary ordered vector spaces. Oriented maps expose every face, and an oriented affine map $f$ separates $C$ from $D$ when $f(D) \le 0 \le f(C)$ element-wise. This generalization of hyperplane separation recovers mathematical objects that are "local", including the generated face at a point and the normal cone. The kernels of separating oriented maps form a lattice of separating affine flats whose bottom element is the joint supporting subspace (JSS), recently introduced in finite dimensions for qualification-free convex analysis results. We define the JSS in general vector spaces for two convex sets $C, D$ as the affine slice of the cross-lineality $\mathrm{lin}(K-G)$, where $K,G$ are the homogenization cones of $C, D$. We show that in finite dimensions, bilateral facial reduction corresponds to a descent in the lattice of separating affine flats, which we formalize as lexicographic products of oriented maps. The lexicographic characterization of faces follows as a corollary. By making no assumption other than convexity, our approach yields general results: we show that the face of a difference of two convex sets is a difference of faces, dropping assumptions of compactness, finite dimensionality, and exposedness. We also show that any face of an intersection of two convex sets is an intersection of faces, resolving open problem 5.10 by Weis in "A note on faces of convex sets", dropping the assumption of non-empty intrinsic core.

We show that the generalized curvatures and more specifically the normal cycle of a compact subset of $\mathbb{R}^d$ are continuous under weak notions of convergence, assuming an a priori mass bound. In particular, provided its mass remains bounded, the normal cycle behaves continuously when the subset is perturbed by a homeomorphism that is $C^0$-close to the identity. Our approach relies on a combination of persistent homology with the geometric measure theory framework classically used in the study of curvatures of singular sets. As an application, we prove that every compact definable set in an o-minimal structure admits a normal cycle by showing that any such set is a limit, in the above sense, of a family of smooth sets whose normal cycles have uniformly bounded mass. We also show that WDC sets are limits of nested smooth sets with uniformly bounded normal cycle masses.

Sliced optimal transport replaces a transport problem in Rd by one-dimensional problems along scalar slices, and has been used both to lower bound the Wasserstein distance W1 (max-sliced distances) and, more recently, to upper bound it by lifting a one-dimensional optimal plan back to the ambient supports (min-sliced plans). We show that for 1-Lipschitz slicers these two directions are two sides of a single sandwich inequality and we characterize when it collapses to equality. If the slicer class contains an optimal Kantorovich-Rubinstein potential, then the same slicer simultaneously maximizes the max-sliced lower objective and minimizes the min-sliced upper objective, and both equal W1. We define the upper objective through a tie-broken transport problem, which removes the dependence on the one-dimensional ordering and yields the identities without any uniqueness assumption on the slice-optimal coupling. The result requires the slicer to be nonlinear in general, consistent with the known failure of exact linear max-sliced/W1 equivalence in dimension greater than or equal to 2, and identifies Kantorovich-Rubinstein potentials as optimal generalized slicers.

Let \(T_\Lambda=\mathbb R^d/\Lambda\), \(d\geq2\), be a flat torus with quotient metric \(\rho_\Lambda\), and consider the distance energies \(I_\alpha(\mu)=\iint \rho_\Lambda(x,y)^\alpha\,d\mu(x)\,d\mu(y)\) of Borel probability measures \(\mu\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(\alpha>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(\rho_\Lambda^\alpha\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(\alpha=2\): for \(1\leq\alpha<2\) they are the translated uniform measures on the two-torsion subgroup; at \(\alpha=2\) all balanced couplings on translates of that subgroup are extremal; and for \(\alpha>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(\alpha\geq1\).

We prove a necessary and sufficient condition for an abstract polyhedron $\Pi$ to be realized as a right-angled, ideal polyhedron in hyperbolic space.