This is about existence for both the max-min angle and the min-max angle problem. Existence of the optimiser is not immediately obvious for both, since the objective functions are combinations of angle functions $$X\mapsto \text{ang}(AXB):=\arccos \frac{(A-X)\cdot(B-X)}{\|A-X\|\|B-X\|}\in[0,\pi]$$ (for given points $A,B$ in the Euclidean plane), which are not defined for $X=A$ or $X=B$, nor have a continuous extension at these points, for in fact they have full oscillation there, equal to $\pi$.

Given a closed $n$-gon $C$ with vertices $V_0,V_1,\dots,V_n=V_0$ listed in cyclic order, a natural definition, for each $i$, is

$$\text{ang}(V_{i-1}V_iV_i)=\text{ang}(V_iV_iV_{i+1}):=\pi-\frac12\text{ang}(V_{i-1}V_iV_{i+1}).$$
(Pictorially, when $X\in C$ coincides with a vertex $V_i$, we may regard the segment $ XV_i$ as an outward directed infinitesimal segment of the bisectrix of the angle at $V_i$).

This way $\sum_{j=0}^{n-1} \text{ang}(V_jXV_{j+1})=2\pi$ for all $X\in C$, so the function
$$C\ni X\mapsto \text{ang}(V_{i-1}XV_i)+\text{ang}(V_iXV_{i+1})=2\pi-\sum_{{j=0}\atop{j\neq i-1,j\neq i}}^{n-1} \text{ang}(V_jXV_{j+1}) $$
is continuous at $X=V_i$, and the function
$$C\ni X\mapsto \big|\text{ang}(V_{i-1}XV_i)-\text{ang}(V_iXV_{i+1})\big|$$
is lower semi-continuous at $X=V_i$, for $$\liminf_{{X\to V_i}\atop {X\in C}} \big|\text{ang}(V_{i-1}XV_i)-\text{ang}(V_iXV_{i+1})\big|\ge\big|\text{ang}(V_{i-1}V_iV_i)-\text{ang}(V_iV_iV_{i+1})\big|=0.$$

Therefore the function

$$C\ni X\mapsto \max\big\{\text{ang}(V_{i-1}XV_i),\text{ang}(V_iXV_{i+1})\big\}=$$$$=\frac12\Big(\text{ang}(V_{i-1}XV_i)+\text{ang}(V_iXV_{i+1})+\big|\text{ang}(V_{i-1}XV_i)-\text{ang}(V_iXV_{i+1})\big|\Big)$$
is also lower semi-continuous at $X=V_i$ and the function
$$C\ni X\mapsto \min\big\{\text{ang}(V_{i-1}XV_i),\text{ang}(V_iXV_{i+1})\big\}=$$$$=\frac12\Big(\text{ang}(V_{i-1}XV_i)+\text{ang}(V_iXV_{i+1})-\big|\text{ang}(V_{i-1}XV_i)-\text{ang}(V_iXV_{i+1})\big|\Big)$$
is upper semi-continuous at $X=V_i$. By consequence, the function $\alpha_*:C\to[0,2\pi]$ defined by

$$\alpha_*(X):=\min_{0\le j\le n-1}\text{ang}(V_jXV_{j+1})=$$$$
=\min\big\{\text{ang}(V_{i-1}XV_i),\text{ang}(V_iXV_{i+1})\big\}\wedge\min_{{0\le j\le n-1}\atop{j\neq i-1,j\neq i}} \text{ang}(V_jXV_{j+1})$$
is upper semi-continuous on $C$, and the function $\alpha^*:C\to[0,2\pi]$ defined by
$$\alpha^*(X):=\max_{0\le j\le n-1}\text{ang}(V_jXV_{j+1})=$$$$
=\max\big\{\text{ang}(V_{i-1}XV_i),\text{ang}(V_iXV_{i+1})\big\}\vee\max_{{0\le j\le n-1}\atop{j\neq i-1,j\neq i}} \text{ang}(V_jXV_{j+1})$$
is lower semicontinuous on $C$.
Hence, the USC function $\alpha_*$ attains its maximum on $C$
$$\theta_*:=\max_{X\in C}\min_{0\le j\le n-1}\text{ang}(V_jXV_{j-1}),$$
and every maximiser is interior to $C$ iff $$\max_{0\le i\le n-1}\min_{0\le j\le n-1}\text{ang}(V_jV_iV_{j+1})<\theta_*;$$
the LSC function $\alpha^*$ attains its minimum on $C$
$$\theta^*:=\min_{X\in C}\max_{0\le j\le n-1}\text{ang}(V_jXV_{j+1}).$$
and every minimiser is interior to $C$ iff $$\min_{0\le i\le n-1}\max_{0\le j\le n-1}\text{ang}(V_jV_iV_{j+1})>\theta^*.$$
(*Rmk*: The next step should be writing these conditions in a simpler equivalent form).

Incidentally, note that from the above identity $\sum_{j=0}^{n-1} \text{ang}(V_jXV_{j+1})=2\pi$ one has $\theta_*\le \frac{2\pi}n\le \theta^*$.

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