(1 \leq k \leq K\)</span>,</p>
<p><span class="math display">\[
x_k &gt; 0,
\]</span></p>
<p>and</p>
<p><span class="math display">\[
\sum_{k=1}^K x_k = 1.
\]</span></p>
<p>An alternative definition is to take the convex closure of the
vertices. For instance, in 2-dimensions, the simplex vertices are the
extreme values <span class="math inline">\((0,1)\)</span>, and <span class="math inline">\((1,0)\)</span> and the unit 2-simplex is the line
connecting these two points; values such as <span class="math inl