# Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $$\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad \|S-\big(Y^TY\big)^{1/2}\|_F\leq\epsilon c^*\}$$ is the biggest set by construction (according to $\epsilon$) included in $$\Delta_1=\{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad S=\Big((Y-AX)^T(Y-AX)\Big)^{1/2}, \|A\|_1\leq \epsilon\}$$ where $(\cdot)^{1/2}$ is the principal square root operator, $\mathbb{S}_{m,m}$ is the set of the symmetric positive semi-definite matrices, $X\in\cal M_{n,m}$ and $Y\in\cal M_{n,m}$ are given matrices, $A\in\cal M_{n,n}$ is a variable matrix and $\|\cdot\|_1$ is the summation of the absolute value of the entries of a matrix. The small term $\epsilon$ is given and the other constant small term $c^*$ is known too where $c^*\leq \frac{1}{m\|V\|\|X\|}$ and $V=Y\big(Y^TY\big)^{-1/2}$ to ensure that $\exists~A$ such that $\|A\|_1\leq\epsilon$ when $S\in\Delta_2$ and hence $S\in\Delta_1$. Thus, by this construction it's clear here that $\Delta_2\subseteq\Delta_1$.

In other words, I would like to prove that $\exists ~\delta$ where $\epsilon_0=\epsilon+\delta$ such that we can find $$S_2\in\Delta_2'= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad \|S-\big(Y^TY\big)^{1/2}\|_F\leq\epsilon_0 c^*\}$$ and $S_2\notin \Delta_1$.

My second question is how to compute the distance in $\|\cdot\|_p$ from the center of the circular set $\Delta_2$ which is $\big(Y^TY\big)^{1/2}$ to the furthest edge of $\Delta_1$.