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Now the answer is almost complete: modulo some extra work on the strictness of relevant inequalities, we do have the uniqueness. The additional ideas used to come to this conclusion are these: (i) to …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{ …

Such a parametrization will not be in general real analytic, because the angle of the support line may be varying too slowly at some points. E.g., let the boundary of the convex set $D$ be $$C:=\{(x, …

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$We have $$\mu(\u+t\v+K)=f(t):=\int_{\R^n}d\x\,F(\x,t),$$ where $$F(\x,t):=\ga(\x)\ …

$\newcommand\de\delta\newcommand\R{\mathbb R}$As in the previous answer, assume that $C$ is a compact convex subset of $\R^2$. The condition that all the width bisectors of $C$ are concurrent means th …

$\newcommand\R{\mathbb R}$It suffices that $f$ just be strongly concave (your conditions 1 and 2 then hold automatically). Indeed, then $$f(y)\le f(x)+h\cdot(y-x)-m|y-x|^2/2$$ for some $x\in\R^2$, som …

We should assume that $C$ is closed; otherwise, there are easy counterexamples. If $C$ is unbounded, then there is at most one width bisector, and hence easy counterexamples again. So, assume that $C$ …

For finite-dimensional $V$, your definition of a face is equivalent to the definition of a poonem, according to part (i) of Exercise 7 on page 21 of the book Convex Polytopes by B. Gruenbaum; then the …

$\newcommand{\R}{\mathbb R} \newcommand{\dd}{\operatorname{d}\!}$ This is not a complete answer, but it may lead to one. Since the volume of the unit $\ell_p$ ball $B_p^d$ is known in closed form, we …

A counterexample is $c_1= 2,c_2= 2,c_3= 81,x_1= -8,x_2= -1,x_3= \frac{2}{9},y_1= 0,y_2= 0,y_3= 0$, $z_1= 0,z_2= 0,z_3= 0$. Added in response to the OP's comment: With the additional condition that t …

$\newcommand{\tb}{\tilde b}\newcommand{\bb}{\bar b}\newcommand{\S}{\mathcal S}\newcommand{\B}{\mathcal B}\newcommand{\T}{\mathcal T}$Note that any totally-monotone function is nonnegative and nonicrea …

No. E.g., let $n=2$ and $$C=\{(s,t)\in\mathbb R^2\colon t\ge s^2\}.$$ Then the asymptotic cone of $C$ is $K:=\{(0,t)\colon t\ge0\}$. Let $u=(1,0)\notin K$. Then $$p((0,t))=(\sqrt t,t)$$ for real $t\ge …

$\newcommand{\ep}{\varepsilon}\newcommand{\R}{\mathbb R}\newcommand{\epi}{\operatorname{epi}}$Yes, $p$ is Lipschitz. Indeed, \begin{equation*} p(x)=x+f(x)u \end{equation*} for $x\in C$, where \beg …

A counterexample is given by the following conditions: $n=185$, $$a_k=\sum_{j=k}^n b_j,\quad b_j:=\frac{c_j}{j+1}, \quad c_j:=\frac{1000}7\,1(j=47)+\frac{302}7\,1(j=185).$$ Indeed, then $a_0\ge\cdots …

Conditions in terms of the function $f$ that are simultaneously necessary and sufficient for such convergence are given in this paper and in its arXiv version. The form of those conditions depends on …