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A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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If $\varphi$ is a smooth norm on $\mathbf{R}^n$ and $A \in \operatorname{End}(\mathbf{R}^n)$ we say that $A$ is accretive with respect to $\varphi$ if $$ \operatorname{grad} \varphi(x) \bullet Ax \ge ...
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Let $K \subsetneq \mathbb{R}^n$ be a compact convex body, $B^n_r$ be the closed $n$-dimensional Euclidean ball of radius $r$, $\rho^{+}(K)$ be the circumradius of $K$, and $\rho^{-}(K)$ be the ...
Peter El Ghazal's user avatar
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Let $(E_n)_n$ be any finite collection of centred ellipses in $\mathbb{R}^2$. Suppose that $E_n$ are pairwise non-homothetic (i.e. there is no positive constant $c>0$ such that $E_n = c E_m$). Now ...
Muduri's user avatar
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I'd like a quick reference for the following fact: Let $(p_1, p_2, \ldots, p_n)$ and $(q_1, q_2, \ldots, q_n)$ be two convex $n$-gons in $\mathbb{R}^2$, both oriented clockwise. Then there are ...
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I recently tried to see in how far polyhedral geometry can be reduced to the study of convex sets with finitely many faces. In other words, I tried to replace "finitely generated" by "...
M. Winter's user avatar
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Let $X$ be a compact convex subset of the plane. Let $c_1$, $c_2$ and $c_3$ be non-collinear points in the interior of $X$. For every point $x$ on the boundary $\partial X$, you can draw the line from ...
Tom Leinster's user avatar
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Let $\Omega_1,\Omega_2\subset\mathbb{R}^n$ be strictly convex bounded domains, possibly with smooth boundary, and such that $\bar\Omega_1\subset\Omega_2$. Let $u\colon \bar\Omega_1\to \mathbb{R}$ be a ...
asv's user avatar
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Consider a non-empty, closed, convex set $C\subseteq\mathbb R^d$ ($d\geq 1$) containing the origin and let $\pi$ be the metric projection onto $C$. Fix some $x\in\mathbb R^d$. Let $f(t)=\|\pi(tx)\|_2$ ...
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I am having a hard time reconciling two facts about Kolmogorov widths and entropy numbers of a given convex body $K \subset \mathbb{R}^n$ with respect to the Euclidean distance. Let $K = B_{1}^n = \{\...
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Let $K\subset \mathbb{R}^n$ be a convex body. We uniformly choose two points $X,Y$ in $K$ and denote the direction of $X-Y$ as $u$, where $u\in \mathbb{S}^{n-1}$, and $f(u)$ is the density of $u$. We ...
ruihan xu's user avatar
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Suppose that $\Omega_1, \Omega_2 \subseteq \mathbb R^n$ are convex domains. We assume that they contain the origin. Then the radial projection $P : \partial\Omega_1 \rightarrow \partial\Omega_2$ ...
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Before posting my question, I would like to mention that I tried posting this at math.stackexchange previously (see here) but I didn't get an answer and I think here is a more appropriate place for ...
spacetimewarp's user avatar
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Let $K\subseteq\mathbb R^d$ be a non-empty, closed, convex set and $x_0\in \partial K$ (the boundary of $K$). Let $u\in\mathbb R^d$ be a vector satisfying $\langle u,x-x_0\rangle\leq 0$ for all $x\in ...
TrivialPursuit's user avatar
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Let $M$ be a 3-dimensional Cartan–Hadamard manifold (complete, simply connected, nonpositive sectional curvature), let $S \subset M$ be a $C^{1,1}$ surface enclosing a domain $E$, and let $D_0$ be the ...
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Let $B$ denote the unit Euclidean ball centered at $0 \in \mathbb{R}^n$. Given a set $K \subset \mathbb{R}^n$ let us denote by $P_K(x)$ the maximal number of points $y_i \in K \cap (x + B)$ such that ...
Drew Brady's user avatar
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Consider a random compact convex set $H\subset\mathbb{R}^d$. I can show that, for any given direction $\theta\in\mathbb{S}^{d-1}$, with probability 1, the $\theta$-exposed face (the set of points $x$ ...
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Let $P \subset \mathbb{R}^n$ be an $n$-dimensional parallelotope generated by $n$ linearly independent vectors $v_1, v_2, \dots, v_n$. For $m \leq n$, I define its $m$-measure $\mu_m(P)$ as the sum of ...
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My question is about finding an optimal definition of the fan of a normal, toric variety. I am writing my dissertation and need to make sure all definitions are correct and optimal. The following are ...
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$\newcommand{\vol}{\mathrm{vol}_n}\newcommand{\T}{\mathsf{T}}$A famous result of Paouris says that if $K$ is a convex body in $\mathbb{R}^n$, then for a universal constant $c_1 > 0$, it holds for ...
Drew Brady's user avatar
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Given a planar compact convex set $C$ and a number $n$, let us try to put $n$ points in $C$ such that the arithmetic mean of the $n\choose 2$ distances between them is to be maximized. Does this ...
Nandakumar R's user avatar
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$\newcommand{\vol}{\mathrm{vol}}$ Let $P_N$ denote the symmetric convex hull of $x_1, \dots, x_N$ which have at most unit norm in $\mathbb{R}^n$. I have seen the following fact used: the volume radius ...
Drew Brady's user avatar
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Let $K \subset \mathbb R^n$ be a symmetric convex body with the following property: if $a,b \in \mathbb Z^n$ are distinct, then $(K+a) \cap (K+b)$ intersect in a set of zero Lebesgue measure. In other ...
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Let $K_N = \mathrm{conv}\{X_1, X_2, \ldots, X_N\}$ be a random polytope in $\mathbb{R}^d$ with $N$ vectors chosen independently and uniformly from the unit ball $B_2^d$. Intuitively, it would make ...
Brayden's user avatar
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Let $\Omega \subset \mathbb{R}^n_x \times \mathbb{R}_t$ be a convex domain. For each $t \in \mathbb{R}$, define the slice $$ \Omega_t := \{ x \in \mathbb{R}^n \mid (x, t) \in \Omega \}, $$ and let $...
complex variable's user avatar
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Let $V_n^d(\mathbb{Q})$ denote the set of realizable volume polynomials of degree $d$ in $n$ variables over $\mathbb{Q}$ - these are polynomials of the form $\frac{1}{d!}\int_Y (\sum x_i D_i)^d$ where ...
DimensionalBeing's user avatar
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Let $X \in \mathbb{R}^{n \times N}$ be a random matrix with columns $X_1, \dots, X_N \sim N(0, I_n)$, independently. Define the minimum $\ell_2^n \to \ell_\infty^N$ singular value $$ s_{N, n} = \inf_{...
Drew Brady's user avatar
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Ref: On special points within convex solids with all planar sections passing through them having equal area. Definition: Given any convex solid C. For any 2 distinct points on its surface, we have a ...
Nandakumar R's user avatar
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Adding a bit to this question: Regular polygon shadows of convex polyhedra Can one construct a convex polyhedron that has planar sections that are equilateral triangle, square and regular pentagon? ...
Nandakumar R's user avatar
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Let $P \subset \mathbb{R}^3$ be a convex regular polytope with a finite group of isometries $G \leq \mathrm{Isom}^+(\mathbb{R}^3)$ acting on it. Suppose $P$ admits at least one pair of antipodal ...
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We define the simple fusion of a finite-supported measure as follows (similar to Elton and Hills 1998): Suppose $P$ and $Q$ are probability measures in $\mathbb{R}^d$ with finite supports $\mathrm{...
Zach Liu's user avatar
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Here are two similar problems: Suppose given a finite subset $V\subset\mathbb{R}^n$. Find an abstract simplicial complex $K$ with vertex set $V$ such that the evident map from $|K|$ to the convex ...
Neil Strickland's user avatar
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Let $X$ be a Banach space with unit ball $B_X:=\{x \in X \colon \|x\| \le 1\}$. A face $F \subset B_X$ is a non-empty convex subset such that if $tx + (1-t)y \in F$ for some $x,y \in B_X$ and any $t \...
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Let $\Gamma$ be a $C^1$ closed convex surface in a non-positively curved $3$-manifold. Suppose that $\Gamma$ has non-negative curvature in the sense of Alexandrov (i.e., in terms of triangle ...
Mohammad Ghomi's user avatar
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In d dimensional space, one can put d+1 nonoverlapping unit balls such that they form a clique, i.e., pairwise touch each other. Can one say that for any d, there is no d-dimensional solid body such ...
Nandakumar R's user avatar
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For now just speaking in $\mathbb{R}^2$ we call a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ positively homogeneous if for every $\lambda \ge 0$ we have $f(\lambda x )=\lambda f(x)$. For such a ...
amr's user avatar
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Many definitions of intrinsic volumes I have searched so far looked rather formulaic and less inspiring. After struggling with the meaning of intrinsic volumes for a while, and inspired by papers like ...
gamja's user avatar
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4 votes
1 answer
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Let $P\subset \mathbb{R}^n$ be a convex compact polytope. Let $\mathbb{R}^n =L\oplus L^\perp$ be an orthogonal decomposition. For any $x\in L$ consider the polytope $$P(x):=P\cap (x+ L^\perp).$$ ...
asv's user avatar
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I am studying the following function but I have trouble checking if it is convex or not... For $N>1$, we let $A$ be a $N$ by $N$ matrix with positive coefficients. For $X \in \mathbb{R}^N$ with ...
Anthony's user avatar
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15 votes
2 answers
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Studying functional analysis, I have accidentally formulated the following statement: Claim. Suppose that $(X, \Vert \cdot \Vert)$ is a normed vector space (over the field of real or complex numbers) ...
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Let $U \subset \mathbb{R}^n$ be an open set. Let $\Phi : \bar{U} \to \bar{\Phi(U)}$ be a $C^1$ diffeomorphism defined up to the boundary (i.e., $\Phi$ is a diffeomorphism from an open set containing $\...
Mathguest's user avatar
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Let $f : \mathbb{R}^n \to \mathbb{R}$ be a locally Lipschitz function, and let $\bar{x} \in \mathbb{R}^n$. Assume the directional derivative $$ f'(\bar{x}; v) := \lim_{t \downarrow 0} \frac{f(\bar{x} +...
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Crossposted at Mathematics SE Let us define the following class of functions: $$ \mathcal{F} := \left\{ f : \mathbb{R}^d \to \mathbb{R} \,\middle|\, f(x) = \sum_{i=1}^n \varphi_i(\xi_i^\top x),\ \...
Ernest's user avatar
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Let $B^n_\infty = [-1, 1]^n$. Let $E \sim \nu_{n,k}$ be a uniformly random $k$-dimensional subspace of $\mathbb{R}^n$. Set the spherical width of $E \cap B^n_\infty$ to be $$ w(E \cap B^n_\infty) = \...
Drew Brady's user avatar
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7 votes
1 answer
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Let $B$ and $S:=\partial B$ denote, respectively, the unit ball and the unit sphere wrt to some norm $\|\cdot\|$ on $\Bbb R^2$. For a fixed real $r\in(0,1]$ and all $u\in S$, let $$D_u:=(B+ru)\cap(B-...
Iosif Pinelis's user avatar
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2 votes
1 answer
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Let $X \equiv (X, \|\cdot\|)$ be a real Banach space. Fix $t \in (0, 2]$. The modulus of convexity for $X$ is $$ \delta(t) = \inf_{x, y \in X} \Big\{\, 1 - \Big\|\frac{x + y}{2}\Big\| : \|x\| \leq 1, ...
Drew Brady's user avatar
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Suppose you are given a given $r$ linear inequalities in $\mathbb{R}^d$ whose intersection defines a convex polytope $E$. Let the upper bounding set $\bar{E}$ be the smallest set of points such that ...
user558301's user avatar
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2 votes
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Let $K$ be a convex body in $\mathbb{R}^n$ and suppose $V(K) = 1$. I am interested in maximizing/minimizing the quantity $$V_1(K, -K) = \lim_{\epsilon \to 0} \frac{V(K -\epsilon K) - V(K)}{\epsilon}$$ ...
Brayden's user avatar
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Let $K \subset \mathbb{R}^n$ denote a centrally symmetric convex body, with corresponding norm $\|\cdot\|_K$. We define the parameters $$ M(K) = \mathbb{E} [\|x\|_K] \quad \mbox{and} \quad b(K) = \...
Drew Brady's user avatar
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In a previous question, I asked whether $(0,0)$ need to be in the convex hull generated the by common points of the quadratic curves $$ \begin{align} g_1&:=x y-a-b x-c y+a x^2=0 \\ g_2&:=y^2-...
Tiago Verissimo's user avatar
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1 vote
2 answers
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I have the following polynomials in $\mathbb{R}[x,y]$ $$ \begin{aligned} g_1 &= xy - a - b x - c y +a x^2 \\ g_2 &= y^2 - h - f x - g y + (h-1) x^2 \end{aligned} $$ where $a,b,c,f,g,h \in \...
Tiago Verissimo's user avatar
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