Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation

This setting fixes the standing notation of differential calculus for real-valued functions on open subsets of Euclidean space, together with the notions of semicontinuity, local extrema, convexity and semiconvexity that accompany it. It introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Background notation)¶ Throughout, $q$ denotes a natural number with $1\le q$, and the notation of clauses 2 to 5 is introduced for every such $q$ simultaneously; a result adopting this setting uses it for whichever dimensions it names. The notation of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation is in force, namely the real numbers with their order and absolute value $|\cdot|$, the terms positive and nonnegative, the natural numbers, the initial segments $[q]$, sequences and their convergence, and Euclidean space $\mathbb{R}^{q}$ with the sum $x+y$, the scalar multiple $\mu x$, the difference $x-y$, the dot product $x\cdot y$, the Euclidean norm $\lVert\,\cdot\,\rVert$, the Euclidean distance $d_{E}$, the origin $0_{\mathbb{R}^{q}}$ and the notions of openness, closedness, boundedness and compactness, the real matrices with their sums, differences, scalar multiples and products, the transpose, the matrix-vector product $Bh$ and the identity matrices, the set $\mathcal{S}(q)$ of symmetric real $q\times q$ matrices, the positive semidefinite ordering $\preceq$, and the norm $\lVert B\rVert$ and distance $d_{\mathcal{S}(q)}$ on $\mathcal{S}(q)$.

2. (Partial derivatives, class $C^{k}$, gradients and Hessians)¶ Let $V\subseteq\mathbb{R}^{q}$ be open and let $f:V\to\mathbb{R}$. For $i\in[q]$ and $x\in V$, $\partial_{i}f(x)$, also written $\frac{\partial f}{\partial x_{i}}(x)$, is the partial derivative of $f$ with respect to the $i$th variable at $x$, and $\partial_{l}\partial_{i}f$ is the iterated partial derivative of clause 4 of C^k Maps on a Euclidean Open Set. For a natural number $k$, that $f$ is of class $C^{k}$ on $V$ is understood in the sense of clause 3 of the definition of $C^{k}$ maps on a Euclidean open set, and unwinds through clauses 1 and 2 there; in particular, by clause 1 there read through the scalar convention of clause 3, a function of class $C^{1}$ on $V$ is continuous at every point of $V$. For $f$ of class $C^{1}$ we write $Df(x)\in\mathbb{R}^{q}$ for the gradient of $f$ at $x$, and for $f$ of class $C^{2}$ we write $D^{2}f(x)$ for the Hessian matrix of $f$ at $x$, which lies in $\mathcal{S}(q)$ by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian. If $W\subseteq V$ is open, the restriction of $f$ to $W$ is again of class $C^{k}$ on $W$ by claim 3 of Restriction of a $C^k$ Map to an Open Subset, and its partial derivatives at points of $W$ agree with those of $f$ by claim 1 there, so its gradient and Hessian at such points, when these are defined, are those of $f$.

3. (Twice differentiability at a point)¶ For $V\subseteq\mathbb{R}^{q}$ open, $f:V\to\mathbb{R}$, $y\in V$, $p\in\mathbb{R}^{q}$ and $B\in\mathcal{S}(q)$, that $f$ is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$ is as defined there, as is the notation $D^{2}f(y)=B$, which is unambiguous because the pair $(p,B)$ is unique when it exists, by A Symmetric Matrix is Determined by its Quadratic Form, and a Second-Order Expansion by its Coefficients §uniqueness. By Basic Properties of Twice Differentiability at a Point §gradient the first-order coefficient is then the gradient $Df(y)$, and by Basic Properties of Twice Differentiability at a Point §c2 a function of class $C^{2}$ on $V$ is twice differentiable at every point of $V$, with first-order coefficient its gradient and Hessian its Hessian matrix; the two readings of $D^{2}f(y)$ therefore agree, and the notation is used for both without further comment.

4. (Continuity, semicontinuity and local extrema)¶ Let $S\subseteq\mathbb{R}^{q}$ and let $f,g:S\to\mathbb{R}$. Continuity of $f$ on $S$ relative to $S$, upper semicontinuity and lower semicontinuity of $f$ on $S$, and local maxima, including strict ones, and local minima of $f$ relative to $S$, are always understood with the ambient metric space $(\mathbb{R}^{q},d_{E})$ of clause 1, and with the metric of The Absolute Value Metric on the Real Line on $\mathbb{R}$. Purely as notation, $f+g$, $f-g$ and $\mu f$, for $\mu\in\mathbb{R}$, denote the functions from $S$ to $\mathbb{R}$ whose values at $x\in S$ are $f(x)+g(x)$, $f(x)-g(x)$ and $\mu\,f(x)$; nothing is asserted about them beyond this reading of the symbols.

5. (Convexity and semiconvexity)¶ A subset $C\subseteq\mathbb{R}^{q}$ is convex as defined there, and for such a $C$ a function $f:C\to\mathbb{R}$ is convex on $C$, or semiconvex on $C$ with constant $\lambda$ for a nonnegative $\lambda\in\mathbb{R}$, as defined there; the latter means that the function $x\mapsto f(x)+\tfrac{\lambda}{2}\lVert x\rVert^{2}$ is convex on $C$, where $\tfrac{s}{2}$ denotes the product of $s$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. By Quadratic Increment Characterisation of Semiconvexity semiconvexity with constant $\lambda$ is equivalent to the quadratic increment inequality

for all $x,y\in C$ and all real $t$ with $0\le t\le1$.