Bound the Euclidean distance between two points on a cone

Question

We have a cone $y^\top D y=0$ ($[y]_1 \geq 0$), where $y \in \mathbb R^{n+1}$, $[y]_1$ is the first element of $y$, and \begin{equation} D=\left[ {\begin{array}{*{20}{c}} 1&{{0_{1 \times n}}}\\ {{0_{n \times 1}}}&{ - {I_n}} \end{array}} \right] . \end{equation} $y^*$ and $\tilde y^*$ are two points on the cone and we can decompose $\tilde y^*-y^*$ into a weighted sum of s set of unit vectors $$ \tilde y^*-y^*=\beta_1 z_1+\beta_2 z_2+\dots+\beta_l z_l+\beta^\bot z^\bot, $$ where $z_i,i=1,\ldots,l$ ($l \in \{1,\ldots,n\}$, i.e., $l$ can vary from $1$ to $n$ ) are orthogonal to each other and we can calculate them beforehand, and $z^\bot$ is orthogonal to $z_i,i=1,\ldots,l$.

Now we can already bound $\beta^\bot z^\bot$ by a constant $\theta$ $$ \|\beta^\bot z^\bot\| \leq \theta. $$ My question is can we bound $\|\tilde y^*-y^*\|$ in this problem?

Thanks for your valuable comments!

Answer 1

The vector $t=\bar y^*-y^*$ can be unbounded even when $z^\perp=0$, $n=1$, and $l=1$. Indeed, if $n=1$ then the cone (which we denote by $C$) is a union of two rays ${(x,x):x\ge 0}$ and ${(x,-x):x\ge 0}$. Graphically it is easy to guess and then to check that $C-C=\{(x,y)\in\Bbb R^2: |x|\le |y|\}$. It follows that for each vector $z^\perp\in\Bbb R^2$ a set $\{t\in C-C: (t,z^\perp)=0\}$ is unbounded.