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Given a closed set $\varnothing\neq E\subset\mathbb{R}^n$, let $\operatorname{Unp}(E)$ be the set if points $x\in\mathbb{R}^n$ for which there is a unique point $y\in E$ nearest to $x$. Clearly $E\sub …

In the time of COVID-19 everything went online including seminars. A good thing about online seminars is that they can be attended by everyone and many seminars that I attended had more than 50 partic …

Let $(X,\mu)$ be a measure space and let $1<p<\infty$. Question. Is the space $L^p(X,\ell^p)$, $$ \lVert f\rVert_p=\Bigl(\int_X\sum_{i=1}^\infty \lvert f_i\rvert^p\, dx\Bigr)^{1/p}, \qquad f=(f_i)_{i …

The following result is Proposition 2.4.3 in [1]: Theorem. Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if there is $r>0$ s …

I cannot find this paper online. Does anyone have a pdf file of it? Zalgaller, V. A. The k-dimensional directions that are singular for a convex body F in Rn. (Russian) Zap. Naučn. Sem. Leningrad. Otd …

I have been teaching Advanced Calculus at the University of Pittsburgh for many years. The course is intended both for advanced undergraduate students and the first year graduate students who have to …

In the non-convex calculus of variations, in the context of non-linear elasticity, the following classes of mappings $u:\Omega\to\mathbb{R}^n$, $\Omega\subset\mathbb{R}^n$, were introduced by John Bal …

Theorem. If $X$ and $Y$ are uniformly convex Banach spaces, then for $1<p<\infty$ the space $$ X\oplus_pY=X\times Y, \qquad \Vert(x,y)\Vert:=(\Vert x\Vert_X^p+\Vert y\Vert_Y^p)^{1/p} $$ is uniformly …

Brown and Gluck [BG] proved in 1964 that the stable homeomorphism conjecture implies the annulus conjecture. Is the proof of this implication difficult? Is there any other place with the proof of thi …

Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f …

Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary. Question. Is it possible to ex …

The classical definition of an approximately differentiable function is as follows: Definition. Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$ …

Is there a simple proof of the fact that: If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$ embedded into $S^3\setminus A$ that such that the circles $A$ and $B$ are link …

Do you know a good reference for the existence and uniqueness of a smooth structure on $3$-manifolds? As far as I understand topological $3$-manifolds admit a unique smooth structure. I could find th …

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an answe …