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Florian Grundbacher

Publications and source records attributed to Florian Grundbacher.

10 recordsLinked to original sources
arXiv2026

Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is sharp to the linear order in $\varepsilon$, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor $\sqrt{n s(K)}$. Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the $L_p$-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the $L_p$-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.

math.MG↗
arXiv2026

Large Signed Sums and the Polarization Constant of Convex Bodies

As a counterpart to classical vector balancing, we study large signed sums of unit vectors in finite-dimensional Minkowski spaces and develop their connection with polarization problems for convex bodies. For every convex body $K\subset\mathbb{R}^d$ containing the origin in its interior and every number $n\geq1$ of vectors, we show that the corresponding large signed sum and polarization constants coincide in a common quantity $μ(K,n)$, which depends only on the symmetric core $K\cap(-K)$. Our main result determines the sharp universal lower bound \[ μ(K,n) \geq μ(B_\infty^d,n) = \frac{1}{n}\left\lceil\frac{n}{d}\right\rceil, \] identifying parallelotopes as global minimizers for every $d$ and $n$. In the opposite direction, we prove general upper bounds that show, in particular, that the Euclidean ball is a maximizer up to an absolute constant factor. We also obtain sharp and nearly sharp results in several special cases, including the planar setting. As $n\to\infty$, we prove that $μ(K,n)$ converges to the Macphail constant of $K$, equivalently to its $1$-absolutely summing constant. Thus, the finite signed-sum problem provides a discrete counterpart of classical Banach space invariants. Combining this connection with sharp results on projection constants, we characterize equality in the corresponding upper bound for the Macphail constant in terms of maximal real equiangular tight frames. Finally, our methods extend to arbitrary vector families, support functions of compact convex sets, and circumradii and diameters of Minkowski sums.

math.MG↗
arXiv2026

$p$-Means of Convex Bodies: Sharpening Relations and Structural Properties

We study general $p$-means of convex bodies, extending the classical definitions by W. J. Firey via support and gauge functions to two families ranging over all $p \in [-\infty,\infty]$. For values of $p$ beyond the classical ranges, we show that $p$-means of polytopes are again polytopes, yielding simpler structural descriptions. Using a natural characterization of dilates of convex bodies based on their boundary structure, we characterize the equality cases between the two types of $p$-means for the same $p$-value. Extending recent results on standard mean-symmetrizations of convex bodies, we further establish (in almost all instances tight) inequalities quantifying how well arbitrary $p$-means of convex bodies approximate each other. These bounds lead to characterizations and sharp stability results for the equality cases between $p$-means for different $p$-values. As a corollary, every Minkowski centered convex body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance.

math.MG↗
arXiv2026

Exact Banach-Mazur distances of certain $\ell_p$-sums and cones

We determine certain Banach-Mazur distances involving $\ell_p$-direct sums of finite-dimensional real normed spaces and related cone constructions of convex bodies. Using a recent characterization of the optimal Banach-Mazur position with respect to the Euclidean ball, we derive a closed formula for the distance from $X_1 \oplus_p \cdots \oplus_p X_k$ to Euclidean space in terms of the distances of the spaces $X_i$ to Euclidean space. For $p = 1$ we show that if $d_{BM}(X,\ell_1^n) \leq 3$, then $d_{BM}(X \oplus_1 \ell_1^m, \ell_1^{n+m}) = d_{BM}(X,\ell_1^n)$. Interpreting $\ell_1$-sums geometrically as double cones motivates a study of single cones over arbitrary convex bases, for which we establish an analogous result with the simplex replacing $\ell_1$. We further show that in dimension $3$ the distance between single cones with symmetric bases equals the distance between the bases, and that the same equality holds for double cones over planar symmetric bases in arbitrary dimension, under an additional assumption on the distance of the bases to $\ell_1^2$. As consequences, we obtain an explicit isometric embedding of the $2$-dimensional symmetric Banach-Mazur compactum into the $3$-dimensional (non-symmetric) compactum and lift a recent construction of arbitrarily large equilateral sets in the $2$-dimensional symmetric compactum to all higher dimensions.

math.MG↗
arXiv2026

John-type decompositions for affinely-optimal positions of convex bodies

Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describes$\unicode{x2014}$and in the Euclidean case fully characterizes$\unicode{x2014}$the optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains $r L_1 + c \subseteq K \subseteq R L_2 + d$ for convex bodies $K, L_1, L_2 \subseteq \mathbb{R}^n$, translation vectors $c,d \in \mathbb{R}^n$, and reals $r,R > 0$ such that the ratio $\frac{R}{r}$ cannot be decreased by linearly transforming $K$. We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball.

math.MG↗
arXiv2025

On Certain Extremal Banach-Mazur Distances and Ader's Characterization of Distance Ellipsoids

A classical consequence of the John Ellipsoid Theorem is the upper bound $\sqrt{n}$ on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathbb{R}^n$. Equality is attained for the parallelotope and the cross-polytope. While it is known that they are unique with this property for $n=2$ but not for $n \geq 4$, no proof of the characterization of the three-dimensional equality case seems to have ever been published. We fill this gap by showing that the parallelotope and the cross-polytope are the unique maximizers for $n=3$. Our proof is based on an extension of a characterization of distance ellipsoids due to Ader from $1938$, which predates the John Ellipsoid Theorem. Ader's characterization turns out to provide a decomposition similar to the John decomposition, which leads to a proof of the aforementioned $\sqrt{n}$ estimate that bypasses the concept of volumes and reveals precise information about the equality case. We highlight further consequences of Ader's characterization, including a proof of an unpublished result attributed to Maurey related to the uniqueness of distance ellipsoids. Additionally, we investigate more closely the role of the parallelogram as a maximizer in various problems related to the distance between planar symmetric convex bodies. We establish the stability of the parallelogram as the unique planar symmetric convex body with the maximal distance to the Euclidean disc with the best possible (linear) order. This uniqueness extends to the setting of pairs of planar $1$-symmetric convex bodies, where we show that the maximal possible distance between them is again $\sqrt{2}$, together with a characterization of the equality case involving the parallelogram.

math.MG↗
arXiv2025

The Banks Set and the Bipartisan Set May Be Disjoint

Tournament solutions play an important role within social choice theory and the mathematical social sciences at large. We construct a tournament of order 36 for which the Banks set and the bipartisan set are disjoint. This implies that refinements of the Banks set, such as the minimal extending set and the tournament equilibrium set, can also be disjoint from the bipartisan set.

econ.TH↗
arXiv2025

Minkowski chirality: a measure of reflectional asymmetry of convex bodies

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski chirality'' measures to Banach--Mazur distances and to each other, and prove their continuity with respect to the Hausdorff distance. In the planar case, we determine the reflection axes at which the Minkowski chirality of triangles and parallelograms is attained, and show that $\sqrt{2}$ is a tight upper bound on the chirality in both cases.

math.MG↗
arXiv2024

Tightening Inequalities on Volume Extremal $k$-Ellipsoids Using Asymmetry Measures

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach-Mazur distance.

math.MG↗