Non-zero probability of hitting a convex hull of d + 1 i.i.d. points in R d .Let X 1 , … , X d + 1 be d + 1 i.i.d. random points in R d sampled from a continuous probability μ of density f.Let x 0 ∈ R d . Is it true that almost surely with respect to μ, P ( x 0 ∈ C o n v ( X 1 , … , X d + 1 ) ) &gt; 0 where Conv denotes the convex hull of the points?

Question

Non-zero probability of hitting a convex hull of   d  +  1 i.i.d. points in             R        d  .Let       X    1    ,  …  ,      X          d      +      1       be   d  +  1 i.i.d. random points in             R        d   sampled from a continuous probability μ of density f.Let       x    0    ∈            R        d  . Is it true that almost surely with respect to   μ,       P    (      x    0    ∈      C    o    n    v    (      X    1    ,  …  ,      X          d      +      1        )  )  &amp;gt;  0 where Conv denotes the convex hull of the points?

Alanna Downs · Accepted Answer

Step 1Choose   (  d  +  1  ) balls             B        1    ,  …  ,            B              d      +      1       in             R        d   such that   0  ∈  Conv  ⁡  (      x    1    ,  …  ,      x          d      +      1        ) for any choices of       x    i    ∈            B        i   for   i  =  1  ,  …  ,  d  +  1. For instance, fix a regular d-simplex centered at 0 and replace each vertex with a small ball.Step 2Now let       x    0   be a Lebesgue point of f such that   f  (      x    0    )  &amp;gt;  0. Then it is straightforward to prove that       lim          r      →              0        +                  1                  r        d                    |                              B                i                    |                  ∫                  x        0            +      r                        B                i              f  (  x  )        d    x  =  f  (      x    0    )  &amp;gt;  0for each   i  =  1  ,  …  ,  d  +  1. So if   r  &amp;gt;  0 is sufficiently small, then      P    (      x    0    ∈  Conv  ⁡  (      X    1    ,  …  ,      X          d      +      1        )  )  ≥      P        (          ∩              i        =        1                    d        +        1              {          X      i        ∈          x      0        +    r                  B            i        }    )    &amp;gt;  0.

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

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