Search Results

This paper is contained in the 1-st volume of Aleksandrov's Selected works. This has an English translation: Alexandrov, A. D. Reshetnyak, Yu. G. (ed.) Selected works. Part 1: Selected scientific pape …

This fact is trivial. Your assumptions imply that $$f(x)=ax^m\prod(1+x/x_k),$$ where $a>0$ and $x_k>0$. All zeros $-x_k$ are negative because you assume that $a_k\geq 0$, so there are no positive zero …

Yes. The boundary even has a locally finite Haudsorff $(m-1)$-measure. No. A convex function of 1 variable has increasing derivative, but this derivative can have a dense set of jumps. In general, …

If you have an arbitrary system of linear equations and add to it one equation, the dimension of the space of solutions can decrease by at most 1. Do take any $d$ equations of your hyperplanes, order …

Let me explain first what Anton wrote: we construct piecewise linear function $f$ by on $[0,n]$ by induction in $n$. Suppose that it is already constructed on $[0,n]$. Define the right derivative $D^+ …

The answer is negative. Example: $$\max\{ |y|,|x|-1\}.$$ This is a convex piecewise linear function which is not a convex combination of your "tripods", and not a limit of such combinations.

There are no such $C$ at all, when $n>2$. Indeed your conditions imply $h_\phi=\phi$ on $\partial C$. But $h$ is bounded and continuous on $\partial C$, while when $n\geq 2$ there is always a subharmo …

Take $g_1(x)=x\log^2x$. Properties 1,2 are evidently satisfied, and computation of the second derivative shows that it is negative for $0< x<1/e$. Now rescale: $g(x)=g_1(x/e)$.