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Given $N$ geodesic caps on the normalized unit sphere in $\mathbb{R}^d$, and whose total surface area sums to one, what is the maximal surface area their union can cover? We show that when these caps have equal surface area, as both the dimension $d$ and the number of caps $N$ tend to infinity, the maximum proportion covered approaches $1 - e^{-1} \approx 0.632$. Furthermore, this maximum is achieved by a random partial sphere covering. Our result refines a classical estimate for the covering density of $\mathbb{R}^d$ by Erd\H{o}s, Few, and Rogers (Mathematika, 11(2):171--184, 1964)., Comment: 15 pages

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., Comment: 28 pages, 3 figures

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., Comment: 22 pages

Let $\alpha$ be a given real number. It is shown that for a given $\alpha$-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by $\alpha$-affine minorants with a fixed number of break points. This extends a classical result of Macbeath (1951) from convex bodies to a functional setting., Comment: 11 pages, 1 figure

A Minkowski symmetral of an $\alpha$-concave function is introduced, 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 ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ 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 hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases., Comment: 30 pages

This article is concerned with the problem of placing seven or eight points on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ so that the surface area of the convex hull of the points is maximized. In each case, the solution is given for convex hulls with congruent isosceles or congruent equilateral triangular facets., Comment: 13 pages, 1 table, 15 figures, to appear in Involve, a Journal of Mathematics

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., Comment: 27 pages (revision: expanded Section 2, shorter proof of Thm. 4.2). To appear in Communications in Contemporary Mathematics

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., Comment: 15 pages, 1 figure; to appear in Discrete Mathematics

An asymptotic formula is proved for the expected $T$-functional of the convex hull of independent and identically distributed random points sampled from the Euclidean unit sphere in $\mathbb{R}^n$ according to an arbitrary positive continuous density. As an application, the approximation of the sphere by random polytopes in terms of $L_p$ surface area differences is discussed., Comment: 20 pages

For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad {\rm os}_{\psi}(K):=\inf_{K^\prime\supset K}{\rm as}_{\psi}(K) \] where ${\rm as}_{\varphi}(K)$ and ${\rm as}_{\psi}(K)$ are the $L_\varphi$ and $L_\psi$ affine surface area of $K$, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\rm IS}_{\varphi}$ and ${\rm os}_{\psi}$ are continuous. In our main results, we prove Blaschke-Santal\'o type inequalities and inverse Santal\'o type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Sch\"utt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas., Comment: 24 pages; to appear in Journal of Mathematical Analysis and Applications

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices., Comment: 15 pages, 3 figures, 1 table. To appear in Acta Crystallographica A77 (2021)

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope which is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak's dual Brunn-Minkowski theory., Comment: 44 pages, 1 figure. To appear in International Mathematics Research Notices

Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a positive density function. We also show that the Dirichlet-Voronoi tiling numbers satisfy $ \text{div}_{n-1} = (2\pi e)^{-1}(n+\ln n) + O(1)$, which improves a classical result of Zador by a factor of $o(n)$. In addition, we provide a remarkable open problem which is the natural geometric generalization of the famous and fundamental "balls and bins" problem from probability. This problem is tightly connected to the optimality of random polytopes in high dimensions., Comment: Typos, journal ref

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices in the symmetric surface area deviation.