This problem cropped up in the context of scale-insensitive methods for generating random variables. Let $X=R^n \cup \{\infty\}$. Suppose we consider a set of transforms $\cal{T} …

Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is a …

Background and motivation: Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (thos …

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seem …

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a tw …

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which …

My question was prompted by an earlier MO by @Daniel:     Duality map in strictly convex Banach spaces I will even use his symbol $\phi$ below. Let $B$ be an arbitrary …

Is the following statement true? Assume that $(M^n, g, p)$ is a pointed complete manifold with metric $g$, $Rc(g)\geq 0$, ${s_i}$ is a positive sequence decreasing to $0$, and $(M …

Does every connected metrizable locally path connected topological space X admit a compatible metric d so that (X,d) is a length space? (Recall the metric space (X,d) is a length …

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from w …

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further a …

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a close …

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (t …

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$, and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece of a geodesic. Define $\gamma(w)$ as all the points of $S …

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of …