MathOverflow

Is there an explicit solution for the following problem:
find $f(x)\in C^1\lbrack 0,1\rbrack,\ 0\le f(x),\ \int\limits_0^1f(x)dx=A$,
that minimizes $\int_0^1\sqrt{1+f'(x)^2}dx$

It is well-known that, up to isomorphism, the finite subgroups of $\mathrm{PGL}(2,\mathbb{Q})$ (and hence of $\mathrm{PGL}(2,\mathbb{Z})$) are exactly the cyclic groups $C_1,C_2,C_3,C_4,C_6$ and the dihedral groups $D_2,D_3,D_4,D_6$ (where $D_n$ denotes...

I am wondering if the following properties are satisfied by the weigthed blow-up and weighted projective-space. I have tried finding the answers to these questions in the existing literature, but most references specifically about weigthed blow-ups drift...

Let X be a finite type smooth scheme over $\operatorname{spec} \mathbb{Z}$, and let $i \colon Z \hookrightarrow X$ be a regular closed immersion. Denoting by $1_X$ the unit object in the stable motivic homotopy category $\text{SH}(X)$.
I am wondering if there is any nice description of the object $i^!1_X \in \text{SH}(Z)$. My intuition is that it looks like some twist of the unit object $1_Z$ by the codimension $d$ of $Z$ in $X$, as the cotangent complex of $i$ is concentrated in degree...