Papers

We study a family of questions in convexity in which size does not matter: one records a convex cross-section only up to translation and scaling, so that the data attached to a convex body $B$ and a direction $\rho$ is a path in the compact metric space $\A_{n-1}$ of aligned shapes. The object of interest is the asymptotic behaviour of this path as the cutting hyperplane approaches the last supporting hyperplane, encoded by an invariant $T(B,\rho)$ that we call the tail. We show that tails are always continua, that polyhedral and smooth support points are “boring” (the tail is a point), and that non-boring behaviour forces degenerate contact. We show that cross-section paths are locally rectifiable, that every locally rectifiable path is realisable approximately and a dense class exactly, and that exact realisation fails in general: a second-order obstruction of bounded-turning type produces a rectifiable path that is not a cross-section path. For tails, by contrast, no such restriction survives: every continuum of shapes occurs as a tail, on the nose rather than up to approximation. We construct bodies possessing nearly universal points, at which the renormalised cross-sections approximate every planar (more generally $(n-1)$-dimensional) convex shape arbitrarily well; such points can be made dense in the boundary, with arbitrary prescribed tails at the grafting sites. Every result below has been formally verified in Lean~4. We close with several optimisation questions and a higher-codimension variant.

Put \(n\) identical coins on a table with no two overlapping. Which arrangement makes the perimeter of the convex hull of the cluster as small as possible? Despite its elementary statement, the solution of this problem is known only up to four disks. We produce a calculus for hard disk clusters of arbitrary finite size, providing class criticality conditions, first order descent tests, and second order spectral criteria for perimeter minimisation. A central difficulty is that the perimeter formula changes with the hull combinatorics, while the admissible first order geometry changes with the realised contacts. Our approach is guided by the principle that the realised geometry intrinsically determines both the local form of the functional and the admissible motions. As an application, this article takes the first step beyond four disks by providing a solution for the five disk case. The minimum perimeter is \(10+2\pi\), attained in exactly three realised classes. Two admit perimeter preserving flexes, of dimensions one and two, while the third is rigid modulo rigid motions. The first order theory provides pruning criteria, and the reduced admissible space together with its intrinsic Hessian distinguish rigidity, second order instability, and perimeter flat degeneracy.

For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

In this paper, we obtain new upper and lower bounds on the colorful Helly number and the Tverberg number in an abstract convexity space with Radon number $r$. We prove an upper bound of $(r-1)2^r$ on the colorful Helly number, which is a factor $O(r)$ far from the lower bound, $2^{r-1}-1$. The best previous bound, by Holmsen and Lee (2021), was $r^{r^{\log r}}$. As a consequence, we obtain improved quantitative bounds for fractional Helly numbers, the selection lemma, weak $\varepsilon$-nets, and the $(p,q)$-theorem, in abstract convexity spaces. Furthermore, using the improved colorful Helly bound, we prove that the Tverberg number $r_k$ is at most $O(r^{\lceil \log_2 r \rceil})k$. The best previous bound, by Pálvölgyi (2022), was $r^{r^{r^{\log r}}}k$. We also study the $t$-wise Tverberg number $r_{k,t}$, which is the least $\ell$ for which any $\ell$ points can be divided into $k$ parts such that the convex hulls of any $t$ parts intersect. We prove the optimal bound $r_{k,t} = O_t(kr)$, in any $S_4$ separable space. In the other direction, we construct a separable space in which $r_k = \Theta(r^2 k)$, while $r_{k,2}=\Theta(rk)$. This proves that the weak version of Eckhoff's conjecture, which suggested that $r_k=O(rk)$ in any abstract convexity space, fails even in separable spaces. In addition, this shows that the abstract analogue of Reay's conjecture (1979), suggesting that $r_{k,2}=r_k$ in Euclidean spaces, already fails in separable convexity spaces.

We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $\pi^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = \lambda_n(a)^{1/2}/\pi$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.

We introduce an empirical version of dual volumes within dual Brunn–Minkowski theory. These provide a simple model for computation and analysis under symmetrization, including moment inequalities. As an application, we derive a new proof of the Busemann intersection inequality that relies on counting points in slabs rather than working directly with volumes of slices.

For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness the dimension of $E$ for many definitions of dimension simultaneously, for example in a self-affine set $E$ there is a single convergent sequence that has the same Assouad spectrum or intermediate dimension values as $E$ itself.

The generalized Busemann–Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann–Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $\alpha > 0$, the Hilbert packing number $M_H(G, K; \alpha)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $\alpha$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; \alpha)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $\alpha$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $\alpha > 0$, \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; c\alpha)^2 M_H(K^\circ, G^\circ; c\alpha). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; c\alpha)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.

In this paper, We prove that: (A) If $m\ge 4,$ there exists a Platonic geodesic tree having $m$ edges (geodesic segments), which form $m$ platonic angles with measure $120^{o}$ and $(\frac{m(m-1)}{2}-m)$ platonic angles with measure $90^{o}.$ (Existence of a Fermat Coordinate system) (B) If $m=2,$ there exists a Platonic geodesic tree with three vertices and 2 edges, which meet at $120^{\circ}$ or $150^{\circ}$ or $180^{\circ}$ (Minimal spanning geodesic tree). (C) If $m=3,$ there exists a Platonic tree and the Steiner point is identical with the Fermat point, having three edges, which meet at $120^{\circ}$ (Hexagonal coordinate system). (D) If $m$ is even and $m \ge 4,$ there exists a Platonic geodesic tree having $m$ edges , which form platonic angles with only two distinct measures $90^{o}$ and $180^{\circ}.$ (Existence of a Cartesian coordinate system). (E) If $m\ge 4$, there does not exist a Platonic geodesic tree having $m$ edges , which form only platonic angles with measure $120^{o}$ (Non existence of a Fermat coordinate system). (F) If $m\ge 3,$ there does not exist a Platonic geodesic tree having $m$ edges, which form platonic angles with only one measure $150^{o},$ or two distinct measures $120^{\circ}$ and $150^{\circ}$ or two distinct measures $90^{\circ}$ and $150^{\circ}$ (Non existence of a Fermat coordinate system). A classification of platonic geodesic trees with boundary vertex set $V$ and added vertex set $Q$ in the $K-$Space is given by conditions (A), (B), (C), (D), (E), (F), which characterize each vertex in $V\cup Q.$

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $\alpha_\gamma(n)$ be the largest number for which \[ \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $\lambda\in[0,1]$. In this paper, we prove that \[ \alpha_\gamma(n) =1-\frac{2}{n-1} \frac{\Gamma(\frac n2)^2}{\Gamma(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial–tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.

The classical Blaschke–Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_\pi$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.

The product of two Cartesian quaternions can be written as a quadratic form based on a 4x4 matrix made of the symbols 1,i,j,k where i,j,k are imaginary numbers introduced by Hamilton. We generalized this matrix to non-Cartesian bases, and showed that the matrix is made of the metric and the cross tensors. We called it crossmetric tensor. Its symbols are s,a,b,c; they are the elementary crystallographic quaternions. We determined the crossmetric tensors for the six crystal families. We also showed that any unit crystallographic quaternion can be geometrically represented by an infinity of pairs of oriented planes intersecting along the vectorial component of the quaternion such that the angle between them is the semiangle of the rotation. The composition of quaternions follows the intuitive source-target rule. The elementary quaternions a,b,c are geometrically represented by pair of perpendicular and oriented planes intersecting along the axis a,b,c, respectively. Other complementary quaternions noted a',b',c' were also introduced. They are the pairs of planes ma, mb, mc. The elementary quaternions a,b,c follow Hamilton rules on the squares of imaginary numbers. The complementary quaternions follow Hamilton rules on the bi and tri-products. For Cartesian basis, the quaternions i,j,k appear as a specific case of crystallographic quaternions. They are formed by the pairs of perpendicular faces of the cube mx, my, mz. Since these planes intersect along the x, y and z axis, respectively, the elementary and complementary quaternions are equal, i.e. i=i',j=j',k=k', which explains why the square, bi and tri-product Hamilton rules are satisfied all together with only three quaternions i,j,k.

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.

In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary $X^{\infty}$, which we call ideal-genericity, and the other defined from within the affine building \(X\), which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of $X^\infty$. These conditions yield an algorithmic method for constructing geometric configurations in $X \cup X^{\infty}$ (when they exist) that correspond to non-generic triples of antipodal ideal chambers of $X^{\infty}$. Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types $B_2 = C_2$, $G_2$ and $B_3$.

Given finitely many compact convex bodies in $\R^n$, one seeks translates maximizing the volume of their common intersection. A lazy solver merely translates each body so as to place its centroid at the origin. We prove that the lazy strategy always captures strictly more than $\left(\tfrac{2}{n+1}\right)^n$ of the optimal volume, and that this constant is sharp: families of cones over tangent disks, indexed by finite nets on the sphere, approach it. The infimum is not attained. For two convex bodies in the plane the resulting sharp constant $4/9$ closes a gap open since 1996, when de Berg, Cheong, Devillers, van Kreveld and Teillaud proved that centroid alignment of two convex polygons captures at least $9/25$ of the maximum overlap and exhibited examples capturing only $4/9$. The proof rests on the following identity: for a convex body $K$ with centroid at the origin, the intersection of all centroid-recentered compact convex supersets of $K$ equals $\tfrac{1}{n+1}(K-K)$. We close with a promise-problem variant in which the lazy strategy captures at least $\left(\tfrac{n}{n+1}\right)^n > \tfrac1e$ of the optimum, uniformly in the dimension. All results below have been checked in Lean~4; the one classical input quoted rather than proved is the equality case of the Brunn–Minkowski inequality, which enters only for $n\ge2$.

A classical problem in convex and discrete geometry asks for the convex polyhedron of greatest volume whose vertices are chosen from the unit sphere $\mathbb{S}^2$. For a prescribed number $N$ of vertices, the problem is known only in a small number of cases. In this paper we resolve the next outstanding case, $N=9$. We prove that every convex polyhedron with at most nine vertices on $\mathbb{S}^2$ has volume at most $3\sqrt{2\sqrt{3}-3}$, with equality, up to rotation, precisely for a triaugmented triangular prism of an explicitly determined shape. The proof combines combinatorial and geometric reductions with sharp volume estimates. By a theorem of Berman and Hanes (Mathematische Annalen, 1970), a volume maximizer must be simplicial, reducing the $2,606$ combinatorial types of $9$-vertex polyhedra to $50$. We prove that a maximizer cannot have a trivalent vertex, leaving only five combinatorial types, which are treated using geometric and combinatorial arguments. In particular, we determine the exact maximizer within the triaugmented triangular prism class, and characterize the equality case.

We prove several analytic results on (possibly noncompact) complete singular surfaces having locally bounded integral curvature (in short: BIC surfaces). Regarding these as metric measure spaces with the 2-dimensional Hausdorff measure, we show that these are infinitesimally Hilbertian, locally doubling and satisfy a local Poincaré inequality. In particular, this entails the existence of a jointly Hölder continuous heat kernel for the Cheeger Laplacian. Assuming that the negative part of the curvature measure of a BIC surface satisfies a Dynkin-type condition, we show that the surface is bi-Lipschitz equivalent to a BIC surface with a lower bounded curvature measure, entailing global variants of the aforementioned results.

In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.

We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells–Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space–time contact selection required by the Hamilton–Jacobi argument.

For a rectangular box with edges $a_1,a_2,a_3$, the width in a uniform random direction $u \in S^2$ is $w=\sum_i a_i|u_i|$. Under a bijective change of parameters this is also the projected area of a rectangular parallelepiped, whose distribution was derived by Walters. We give a direct co-area derivation that recasts the law as one global positive-part formula and integrates it to an elementary closed cumulative distribution function. The density is real-analytic except at the edge lengths, face diagonals and space diagonal. We classify every singularity as a corner, square-root fold, superposition or terminal jump and compute its coefficient. A Gaussian representation gives all moments from one generating function. The degree of the $n$th cumulant in $\pi^{-1}$ is exactly $\lfloor n/2 \rfloor$; at every order the mean-normalised cumulant is a polynomial in two scale-invariant combinations of the intrinsic volumes, with explicit formulas given through fifth order. The first three cumulants determine the box, and we characterise the admissible triples. Beyond boxes, intrinsic volumes determine neither width variance nor brightness variance. The sign-pattern representation extends to bodies whose central symmetral is a zonotope, while the cellwise co-area method applies to every full-dimensional polytope.

We introduce the notion of an F-space - a set equipped with a family of generalized pseudometrics and marked points, and construct the categorical Gromov-Hausdorff distance for F-spaces. Based on this, we propose a new definition of the distance between dynamical systems, understood in a broad sense as parameterized families of maps. The main focus is on local dynamics: we prove the preservation of Lyapunov stability and asymptotic stability under limit transitions with respect to the new distance (in the latter case, assuming a common radius of attraction). It is established that the class of stable systems is a closed and nowhere dense subset in the space of compact dynamical systems. Next, we investigate the properties of the Hausdorff map, which induces an F-space structure on the family of subsets. In the final part, we introduce another modified version of the Gromov-Hausdorff distance for dynamical systems and use it to calculate the exact distance between torus translations for a non-resonant vector.

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets, the polytope $P_{6,10}$ attributed to Bugaenko. Together with results of Felikson-Tumarkin ($d \ge 7$) and of Burcroff and Ma-Zheng ($d = 4, 5$), this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets. The proof is computer assisted. Affine Gale duality applied to the complete database of order types on 10 points yields 387 combinatorial types, of which Lannér's classification excludes 83. For the remaining 304, an exhaustive search over Coxeter labellings, with no a priori bound on the dihedral angles, leaves a single realizable Gram matrix. Every rejection is certified in exact arithmetic, and completeness of the search is certified independently by DRAT proofs checked by drat-trim. The same code reproduces the known censuses in dimensions 4 and 5. Code, data and certificates are publicly available.

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

We study dual volume approximation of the Euclidean ball by polytopes with a prescribed number of $k$-dimensional faces. This continues the authors' previous work on intrinsic volume approximation of the ball by polytopes with a fixed number of $k$-faces, and develops the corresponding dual radial theory. Our first main result gives a nonasymptotic lower bound for the volume deficit of an inscribed polytope $P_M\subset B_d$ with at most $M$ $k$-faces, for $0\leq k\leq \lfloor d/2\rfloor$. The estimate has the form \[\operatorname{vol}_d(B_d\setminus P_M) \geq \frac{1-e^{-1}}{2}\kappa_d\min\left\{ 1,\frac{d}{2}\left(\frac{\omega_d}{4\kappa_{d-1}}\right)^{\frac{2}{d-1}} M^{-\frac{2}{d-1}}\right\},\] and, in the vertices case $k=0$, in the large-$M$ regime it recovers the order of the lower bound of Gordon, Reisner and Schütt (J. Approx. Theory, 1997). We also prove the polar counterpart for the mean width excess of circumscribed polytopes with at most $M$ $k$-faces, for $\lceil d/2\rceil-1\leq k\leq d-1$. More generally, using the analytic extension of the dual volume deviations introduced by Besau, Hoehner and Kur (Int. Math. Res. Not., 2021), we obtain nonasymptotic lower bounds for all $q\in\mathbb R$, including $q=0$, in both the inscribed and circumscribed models.

We study a reconstruction problem of planar domains from non-local integral-geometric invariants. We show that a generic polygonal domain, not necessarily convex, is uniquely determined, up to Euclidean isometry, by the interpoint distance distribution (IDD), which, for convex domains, is equivalent to the chord length distribution. Using a boundary representation of the Riesz energy function, we replace the IDD of the domain by an equivalent boundary IDD weighted by the scalar product of the outer unit normals, which we call $\nu$-weighted IDD of the boundary. It enables us to reduce the problem to a one-dimensional problem. By analyzing jumps and blow-up terms of the second and third derivatives of the $\nu$-weighted IDD of the boundary, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. This can be considered as extending Waksman's classical generic reconstruction theorem for convex polygons to non-convex setting, as well as extending generic polygonal reconstruction from directional covariogram data to a one-dimensional invariant in which directional information has been integrated out.

We determine the sharp number and the complete finite spectrum of barycentric hyperplanes through the Tukey median. We further prove that every convex body in $\mathbb R^n$, $n\ge3$, contains an interior point incident with at least five barycentric hyperplanes. In particular, this settles Grünbaum's original conjecture in dimension four.

Problem F16 of Croft, Falconer, and Guy asks for the least worst-case length needed to join prescribed pairs of points by pairwise disjoint planar arcs, when the distance within each pair is at most one. The book suggests that the answer for two pairs is $\sqrt2$. We prove this exactly. The lower bound is a crossing argument in a square. The upper bound follows from a sharp ellipse lemma. As a consequence, the value proposed in the book for three pairs, $(\sqrt3+1)/2$, cannot be correct under the literal formulation, because the constants are nondecreasing in the number of pairs.