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Steven Hoehner

Publications and source records attributed to Steven Hoehner.

At least 19 recordsLinked to original sources
arXiv2026

The maximum volume polytope with nine vertices inscribed in the sphere

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.

math.MG↗
arXiv2026

Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

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 Sch\"utt (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.

math.MG↗
arXiv2026

Blaschke operations on log-concave functions and affine isoperimetric inequalities

We introduce Blaschke addition and homothety operations on log-concave functions and study their affine-geometric consequences. Our starting point is the first variation formula of Falah and Rotem (Calc. Var. and PDE, 2026), which associates to each log-concave function a pair of surface area measures. Using the additivity of these measures, we define a canonical Blaschke sum and Blaschke homothety on the class of log-concave functions, uniquely determined up to translation. We establish the basic algebraic properties of these operations, define the associated Blaschke symmetral, and show that this symmetrization preserves both total mass and the first quermassintegral. We also prove that successive Blaschke symmetrizations converge, after translations, to a radially symmetric log-concave function, which we call the mean Blaschke symmetral. We then relate the canonical theory to projection-type constructions. In particular, we show that the functional projection body arising from the first variation coincides with the projection body of the asymmetric LYZ body, and we derive corresponding intertwining properties. As applications, we prove concavity of the entropy with respect to the canonical Blaschke sum and obtain associated Kneser--S\"uss-type inequalities. We also study a functional version of affine surface area, and prove affine isoperimetric inequalities for log-concave functions. In particular, we obtain a Blaschke-concavity property for the affine surface area and show that it is maximized, under fixed first quermassintegral, by radially symmetric functions.

math.FA↗
arXiv2026

Generalized outer linearizations and extremal properties of rotational epi-symmetrizations

We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer linearization is defined as a convex minorant represented by a general but function-dependent set of slopes, thereby extending classical outer representations of convex bodies by supporting halfspaces. This representation converts geometric outer approximations by supporting halfspaces into functional approximations by supporting affine functions, and replaces outer normal data by a dual sampling problem in the domain of the Legendre--Fenchel transform. On a standard class of coercive convex functions, we derive a general extremal principle, showing that the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence. A central feature of the analysis is that it is carried out in the natural class of coercive, but not necessarily super-coercive, convex functions. Working in this setting introduces intricate topological and variational difficulties, which are addressed using refined duality and epi-convergence arguments. As an application of our main results, we derive a functional version of Urysohn's inequality, as well as an analytic extension of a classical covering result of Firey and Groemer (J. London Math. Soc., 1964). Finally, we prove an extremal inequality related to the piecewise affine approximation of convex functions.

math.FA↗
arXiv2026

Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous geometric analogue of the classical balls-into-bins problem. We establish a Central Limit Theorem for the volume of the resulting random partial covering, showing that its fluctuations are asymptotically Gaussian. Moreover, we obtain a quantitative bound on the rate of convergence in the Kolmogorov distance. Our results hold both in fixed dimension and in a high-dimensional regime where the dimension grows at most logarithmically with $N$.

math.PR↗
arXiv2026

One polytope fits all: Characterization of the Euclidean ball via simultaneous intrinsic volume approximation

We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both the deterministic and probabilistic settings. In the deterministic model of inscribed approximation, we show that if a single sequence of polytopes is asymptotically best for the volume and mean width difference simultaneously, then $K$ must be a Euclidean ball. In particular, the Euclidean ball is the unique $C_+^2$ convex body for which one sequence of polytopes can approximate all intrinsic volumes simultaneously at the optimal asymptotic rate. In the probabilistic model, we prove a stronger statement: if a single sampling density on $\partial K$ yields random inscribed polytopes that are asymptotically optimal (in expectation) for any two distinct intrinsic volume deviations, then $K$ must be a Euclidean ball. Moreover, using polarity, we establish dual versions of this rigidity theorem for polytopes circumscribed about $K$ (with a restricted number of facets) in the volume and mean width cases, again in both deterministic and probabilistic frameworks. The proofs use tools from asymptotic quantization theory together with the curvature-based optimal vertex distributions. These results resolve an open question posed by Besau, Hoehner and Kur ({\it IMRN}, 2021).

math.MG↗
arXiv2025

From Circles to Convex Bodies: Approximating Curved Shapes by Polytopes

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is therefore: how well can a smooth, curved convex body be approximated by a polytope with only $N$ faces? A striking phenomenon is that in $\mathbb{R}^d$, many seemingly different approximation errors--such as volume, surface area, and others) often decay like $N^{-2/(d-1)}$ when the body has smooth, positively curved boundary. This survey article offers a guided tour of that ``universal exponent'', starting from the classical approximation of a circle by an $N$-gon and building intuition via spherical caps and curvature. We then survey a few representative theorems--including results showing that random polytopes can be almost as good as best possible ones--and explain why the Euclidean ball is a natural benchmark for the ``hardest" case. We also highlight a recently introduced projection-based distance that compares bodies through the average distance of their shadows. Finally, we list accessible open problems about sharp constants, dimension dependence, and gaps between known upper and lower bounds.

math.MG↗
arXiv2025

Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. B\"or\"oczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$.

math.MG↗
arXiv2025

New fiber and graph combinations of convex bodies

Three new combinations of convex bodies are introduced and studied: the $L_p$ fiber, $L_p$ chord and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the $L_p$ fiber and $L_p$ chord combinations, we derive Brunn--Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J., 2014) on the monotonicity of the general affine surface areas under Steiner symmetrization. As an application, we deduce a corresponding Minkowski's first inequality for the $L_p$ affine surface area of a graph combination of convex bodies.

math.MG↗
arXiv2025

On the Optimality of Random Partial Sphere Coverings in High Dimensions

Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, whose total normalized surface area is one, what is the maximal proportion of the sphere that their union can cover? In this work, we provide an asymptotically sharp upper bound for an antipodal partial covering of the sphere by $N=N(d)$ congruent caps in the regime $N(d)\to\infty$ and $\ln N(d)=o(\sqrt d)$, showing that the maximum proportion covered approaches $1 - e^{-1}$ as $d\to\infty$. We discuss the relation of this result to the optimality of random polytopes in high dimensions, the limitations of our technique via the Gaussian surface area bounds of K. Ball and F. Nazarov, and its applications in computer science theory.

math.MG↗
arXiv2024

Steiner symmetrization on the sphere

The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by J. Schneider (Manuscripta Math. 60: 437-461, 1988). We show that this symmetrization preserves volume in every dimension, and convexity in the spherical plane, but not in dimensions $n > 2$. In addition, we investigate the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap. We apply our results to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally symmetric spherically convex disks. Lastly, we prove a spherical variant of a theorem of Winternitz.

math.MG↗
arXiv2024

On maximal intersection position for logarithmically concave functions and measures

A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports. As an application of our results, we introduce a John-type position for even $\log$-concave measures.

math.FA↗
arXiv2023

An extremal property of the symmetric decreasing rearrangement

It is shown that for a given log-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by inner log-linearizations with a fixed number of break points. This extends a classical result of Macbeath (1951) from convex bodies to a functional setting.

math.FA↗
arXiv2023

On Minkowski symmetrizations of $\alpha$-concave functions and related applications

The Minkowski symmetral of an $\alpha$-concave function is studied, and some of its fundamental properties are derived. It is shown that for a given $\alpha$-concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``reflectional hypo-symmetrization''. As an application, it is shown that the reflectional hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ is by ``inner log-linearizations'' with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the reflectional hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases.

math.FA↗
arXiv2022

An intrinsic volume metric for the class of convex bodies in $\mathbb{R}^n$

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

math.MG↗
arXiv2022

Extremal arrangements of points on the sphere for weighted cone-volume functionals

Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived, including extremal properties of the regular polytopes involving the $L_p$ surface area. Some applications to crystallography and quantum theory are also presented.

math.MG↗