Non-zero probability of hitting a convex hull of d + 1 i.i.d. points in R...

Step 1
Choose $(d+1)$ balls ${\mathsf{B}}_1,\dots ,{\mathsf{B}}_{d+1}$ in ${\mathbb{R}}^d$ such that $0\in \mathrm{Conv}(x_1,\dots ,x_{d+1})$ for any choices of $x_i\in {\mathsf{B}}_i$ for $i=1,\dots ,d+1$. For instance, fix a regular d-simplex centered at 0 and replace each vertex with a small ball.
Step 2
Now let $x_0$ be a Lebesgue point of f such that $f(x_0)>0$. Then it is straightforward to prove that
$\underset{r\to 0^{+}}{lim}\frac{1}{r^d|{\mathsf{B}}_i|}{\int }_{x_0+r{\mathsf{B}}_i}f(x)\mathrm{d}x=f(x_0)>0$
for each $i=1,\dots ,d+1$. So if $r>0$ is sufficiently small, then
$\mathbb{P}(x_0\in \mathrm{Conv}(X_1,\dots ,X_{d+1}))\ge \mathbb{P}({\cap }_{i=1}^{d+1}\{X_i\in x_0+r{\mathsf{B}}_i\})>0.$