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<math>C^* </math> is always a [[convex cone]], even if <math>C </math> is neither [[convex set|convex]] nor a [[linear cone|cone]].
When <math>C </math> is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=9780521833783|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|pages=51-53}}</ref>
* A non-zero vector <math>y</math> is in <math>C^*</math> if and only if both of the following conditions hold: (i) <math> y </math> is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] <math>C </math>. (ii) <math> y </math> and <math>C </math> lie on the same side of that supporting hyperplane.
*<math>C^* </math> is [[closed set|closed]] and convex.
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