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No, the Loewner ellipsoid is not monotone w.r.t. inclusion. Let $K$ be a square, whose Loewner ellipsoid is its circumcircle. Let $L$ be any other ellipse through the four vertices of $K$. The Loewner …

No. Let $X$ be a tripod (three segments with one common endpoint), $d=1$, $r=2$ and $E$ the set of the 3 leaf points.

This is the second half of a proof started by Peter Shor. I assume that the set of arcs is already in a position as in Peter's answer: the red arcs $(L_1,R_i)$ are cyclically ordered and all blue arc …

No. Consider $f(x,y)=e^x+y^2$, then $\varphi(x,y)=(e^x,2y)/(e^x+y^2)$. The image of $\varphi$ has only one point $(1,0)$ on the axis $y=0$. The points $a:=\varphi(0,1)=(\frac12,1)$ and $b:=\varphi(0,- …

There is a counter-example. Note that in any normed space, its unit ball satisfies your supremum-attaining property. Indeed, for any $x_0\in X$ the supremum of $d(x_0,\cdot)$ on the ball is attained …

No. Consider solid angles in $\mathbb R^2$. They all (except the half-plane) are isomorphic, yet one may be strictly included in another. Furthermore, a generic cone in $\mathbb R^d$, $d\ge 3$, has a …

I'm adding 'convexity' tag because some extremal properties of ellipsoids might be relevant here. …

If the second $\delta(\varepsilon)$ is allowed to differ from the first one, then there is a simple implicit argument: Suppose the contrary, then there is a sequence $X_n$ of 2-dimensional normed spac …

In dimension 1, Radon's theorem says that for any 3 points on the real line, one of them belongs to the segment between the two others. This becomes false if one replaces the real line by the tripod ( …

No, even if you assume that $f$ is convex on $\mathbb R^{d+n}$. Take $n=2$ and let $y_1,\dots,y_{d+1}\in\mathbb R^2$ be vertices of a convex polygon. Let $X_1,\dots,X_{d+1}\subset\mathbb R^n$ be your …

No such a set is not always a polytope. Consider the convex hull of the set of points of the form $(1/n.1/n^2)$ and $(0,0)$ in $\mathbb R^2$. Its boundary is a union of infinitely many segments with r …

The answers are no to Question 1, and yes to Question 2 (assuming $n\ge 2$). Let $h$ be the support function of $K$. The projection of $K$ to a linear subspace $L$ is central symmetric iff the restri …

On the other hand, $h(K)\subset K$ due to convexity. Thus $h(K)\subset K\setminus int(K')$, hence $Vol(h(K))\le Vol(K\Delta K')<\epsilon$. …