A set \(C\subseteq\mathbf R^n\) is affine set if for any two distinct points lie in \(C\), \(x_1,x_2\in C\) and \(\theta\in \mathbf R\), the linear combination of these two points lies in \(C\), \(\theta x_1+(1-\theta)x_2\in C\) with the coefficients sum to one. This kind of linear combination is called affine combination. The set of all affine combinations of points in set \(C\subseteq\mathbf R^n\) is called the affine hull of \(C\), and denoted \[\mathbf{\text{aff }}C=\{\theta_1x_1+\cdots+\theta_kx_k|x_1,\cdots,x_k\in C,\theta_1+\cdots+\theta_k=1\}\] The affine hull is the smallest affine set that contains \(C\).
A set \(C\subseteq\mathbf R^n\) is convex set if for any two distinct points lie in \(C\), \(x_1,x_2\in C\) and \(0\le\theta\le1\), the line segment between these two points lies in \(C\), \(\theta x_1+(1-\theta)x_2\in C\). The set of all convex combinations of points in set \(C\subseteq\mathbf R^n\) is called the convex hull of \(C\), and denoted \[\mathbf{\text{conv }}C=\{\theta_1x_1+\cdots+\theta_kx_k|x_1,\cdots,x_k\in C,\theta_1,\cdots,\theta_k\ge0,\theta_1+\cdots+\theta_k=1\}\] Or using vector notation as \[\mathbf{\text{conv }}C=\{\theta_1x_1+\cdots+\theta_kx_k|x\in C,\theta\succeq0,\mathbf 1^T\theta_=1\}\]
A set \(C\subseteq\mathbf R^n\) is cone set or nonnegative homogeneous if for every point \(x\) in \(C\), \(x\in C\) and \(\theta\ge0\), then we have \(\theta x\in C\). A set \(C\subseteq\mathbf R^n\) is convex cone if for any two distinct points lie in \(C\), \(x_1,x_2\in C\) and \(\theta_1, \theta_2\ge0\), we have \[\theta_1x_1+\theta_2x_2\in C\] A point of the form \(\theta_1x_1+\cdots+\theta_kx_k\) with \(\theta_1,\cdots,\theta_k\ge0\) is called a conic combination. The set of all conic combinations of points in set \(C\subseteq\mathbf R^n\) is called the conic hull of \(C\), and denoted \[\{\theta_1x_1+\cdots+\theta_kx_k|x_1,\cdots,x_k\in C,\theta_1,\cdots,\theta_k\ge0\}\] which is also the smallest convex cone that contains \(C\).
Let \(X\) be a convex set and let \(f:X\to\mathbb R\) be a function, \(f\) is called convex if: \[\forall x_1,x_2\in X, \forall t\in[0,1]:\quad f(tx_1+(1-t)x_2)\leq tf(x_1)+(1-t)f(x_2)\] Suppose \(f\) is a function of one real variable defined on an interval, and let \[R(x_1,x_2)=\frac{f(x_2)-f(x_1)}{x_2-x_1}\] where \(R(x_1,x_2)\) is the slope of the straight line of the two points \(f(x_1)\) and \(f(x_2)\), \(f\) is convex if and only if \(R(x_1,x_2)\) is monotonically non-decreasing in \(x_1\), for every fixed \(x_2\) (or vice versa).
A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its tangents: \[f(y)\ge f(x)+f'(x)(y-x)\] or \[f(x+h)\ge f(x)+f'(x)h\] for all \(x\) and \(y\) in the interval.