Jorge L. Arocha, Javier Bracho, Luis Montejano
We prove that if a convex set in Cn contains two inscribed complex ellipsoid of maximal volume then one is a translate of the other. On the other hand, the circumscribed complex elipsoid of minimal volume is unique. As application we prove the complex analoge of Brunn's characterization of ellipsods.
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Egan conjecture holds
该研究证明了维数不小于2的欧几里得单形满足Egan猜想，确立了其外接球半径、内切球半径与两球心距离间的不等式关系。
Taxicab Triangle Incircles and Circumcircles
研究taxicab几何中的内接角，并应用于证明三角形taxicab内切圆和外接圆的存在性与唯一性。
Given a planar pentagon, construct two new pentagons: the vertices of the first one are the intersection points of the diagonals of the original pentagon, and the vertices of the second one are the tangency points of the conic inscribed in the original pentagon. E. Kasner theorem, published in 1928, asserts that these two operations on pentagons commute. We extend Kasner's result to Poncelet polygons, that is, the polygons inscribed into a conic and circumscribed about a conic.
On a Dowker-type problem for convex disks with almost constant curvature
本文研究Dowker型问题在将Hausdorff拓扑替换为表面积测度拓扑时，关于几乎常曲率凸圆盘的最大内接多边形面积序列凹性是否成立。
On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body
本文通过研究凸体的外接锥特征，证明了在特定条件下，凸体或其子集为椭球的条件。
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rollin…
We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the ellipsoid. Simple examples with non-unique minimal or maximal ellipsoids conclude this article.
In this paper we prove asymptotic upper bounds on the variance of the number of vertices and missed area of inscribed random disc-polygons in smooth convex discs whose boundary is $C^2_+$. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.
Dowker-type theorems for disk-polygons in normed planes
本文研究了赋范平面上由单位圆盘平移交集定义的 $C$-$n$-gons 的 Dowker 型定理，证明了外切 $C$-$n$-gons 的面积和周长及内接 $C$-$n$-gons 的周长定理成立，并发现对于典型原点对称凸体，内接面积定理不成立。
We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, D. We also prove that the smallest nonnegative angle between equal conjugate diameters of E_{I} equals the smallest nonnegative angle between the diagonals of D. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let D be any convex quadrilateral. In previous papers, the author proved that there is a …
Ellipsoids in pseudoconvex domains
研究伪凸域中hermitian椭球体的最大体积问题，证明存在性和唯一性并给出最大化的特征化。
Given an arbitrary convex symmetric n-dimensional body, we construct a natural and non-trivial continuous map which associates ellipsoids to ellipsoids, such that the Lowner-John ellipsoid of the body is its unique fixed point. A new characterization of the Lowner-John ellipsoid is obtained, and we also gain information regarding the contact points of inscribed ellipsoids with the body.
Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.
A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups of convex bodies and their extremal ellipsoids. We show that if the automorphism group of a convex body K is large enough, then it is possible to determine the extremal ellipsoids CE(K) and IE(K) exactly, usin…
A Spectral Proof of Khachiyan's Ellipsoid Conjecture
本文通过谱方法证明了Khachiyan关于最大内接椭球中心切割体积比的猜想，给出最优常数√e/2，并由AI辅助发现、Lean 4验证。
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