correcting typos

doc/src/week39/week39.do.txt

@@ -338,7 +338,16 @@ regular polygons (triangles, rectangles, pentagons, etc...).
 !split
 ===== Convex function =====
 
-_Convex function_: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.
+_Convex function_: Let $X \subset \mathbb{R}^n$ be a convex
+set. Assume that the function $f: X \rightarrow \mathbb{R}$ is
+continuous, then $f$ is said to be convex if $f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2)$
+for all $x_1, x_2 \in X$ and for all $t \in [0,1]$.
+If $\leq$ is replaced with a strict inequaltiy in the
+definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said
+to be strictly convex. For a single variable function, convexity means
+that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the
+value of the function on the interval $[x_1,x_2]$ is always below the
+line as illustrated below.
 
 !split
 ===== Conditions on convex functions =====
@@ -346,13 +355,15 @@ _Convex function_: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the
 In the following we state first and second-order conditions which
 ensures convexity of a function $f$. We write $D_f$ to denote the
 domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more
-details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/, 2004".
+details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/".
 
 !bblock First order condition
 Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for
 all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$
-is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
-for all $x,y \in D_f$. This condition means that for a convex function
+is a convex set and $f(y) \geq f(x) + \nabla f(x)^T (y-x)$ holds
+for all $x,y \in D_f$.
+
+This condition means that for a convex function
 the first order Taylor expansion (right hand side above) at any point
 a global under estimator of the function. To convince yourself you can
 make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and