Proof of Lemma 16. The local semiconcavity in Definition 8 implies that the function \[f(t):=\frac{|p\gamma(t)^2|-|px|^2-\lambda t^2}{2|px|t}\] is nonincreasing in \(t>0\). Hence the limit exists when \(t\to 0\). Furthermore, \(f(t)\) is \(((\lambda-1)t/2|px|)\)-close to the function \[g(t):=-\cos\tilde{\angle} px\gamma(t)=\frac{|p\gamma(t)|^2-|px|^2-t^2}{2|px|t}.\] Thus the limit of \(g(t)\) also exists and coincides with that of \(f(t)\). Together with the monotonicity of \(f(t)\), this implies the desired inequality ?? (cf.Remark 17). See [16] for more details (cf.[15]). ◻

Proof. Let \(\gamma(t)\) (resp.\(\gamma_i(t)\)) denote the shortest path from \(x\) to \(y\) (resp.\(x_i\) to \(y_i\)) parameterized by arc-length. By definition, \(\angle pxy\) is close to \(\tilde{\angle} px\gamma(t)\) for small \(t>0\). For sufficiently large \(i\) depending on \(t\), \(\tilde{\angle} px\gamma(t)\) is close to \(\tilde{\angle} p_ix_i\gamma_i(t)\). Applying the almost comparison inequality ?? to \(\tilde{\angle} p_ix_i\gamma_i(t)\) and taking the limit as \(i\to\infty\) and \(t\to0\), we get the desired inequality. See [16] for more details. ◻

Proof. Let \(\gamma(t)\) be the shortest path from \(y\) to \(z\) such that \(\gamma(0)=x\). By Remark 11, the distance function \(d(p,\cdot)\) is locally semiconcave around \(x\). Hence its directional derivative satisfies \[d(p,\gamma(0))^++d(p,\gamma(0))^-\le 0\] (we use the notation \(d(p,\gamma(0))^\pm\) to represent the directional derivative of \(d(p,\cdot)\) in the direction \(\gamma^\pm(0)\) rather than the right/left derivative). By the first variation formula 1 , this is equivalent to the desired inequality. See [16] for more details (cf.[15]). ◻

Proof. Let \(\gamma(t)\) be the shortest path from \(x\) to \(y\) parameterized by arc-length. By the triangle inequality and the fact that \(p,x,q\) are on a shortest path, we have \[|p\gamma(t)|-|px|+|q\gamma(t)|-|qx|\ge0.\] Dividing both sides by \(t>0\) and taking \(t\to0\) as in the first variation formula 1 , we obtain the desired inequality (note that \(\cos\alpha+\cos\beta\le0\) if and only if \(\alpha+\beta\ge\pi\) for any \(\alpha,\beta\in[0,\pi]\)). ◻

Proof. Let \(\gamma(t)\) be the shortest path from \(x\) to \(y\) parameterized by arc-length. By the triangle inequality and the fact that \(p,p',x\) are on a shortest path, we have \[|p\gamma(t)|-|px|\le|p'\gamma(t)|-|p'x|\] Dividing both sides by \(t>0\) and taking \(t\to0\) as in the first variation formula 1 , we obtain the desired inequality. ◻

Proof. Let \(q\in B\) be a point on the extension of the shortest path \(px\) beyond \(x\). See Figure 3.

By Lemma 24 for opposite angles, we have \[\angle pxy+\angle qxy\ge\pi,\quad \angle pxz+\angle qxz\ge\pi.\] On the other hand, by Lemma 19 for adjacent angles, we have \[\angle pxy+\angle pxz\le\pi,\quad\angle qxy+\angle qxz\le\pi.\] Combining the above four inequalities, we obtain the desired equality. ◻

Proof. Let \(z\in B\) (resp.\(z_i\in B\)) be a point on the extension of the shortest path \(yx\) (resp.\(y_ix_i\)) beyond \(x\) (resp.\(x_i\)). We may assume that \(z_i\) converges to \(z\). By Lemma 27, we have \[\angle pxy+\angle pxz=\pi, \quad \angle p_ix_iy_i+\angle p_ix_iz_i=\pi.\] On the other hand, by Lemma 18, we have \[\liminf_{i\to\infty}\angle p_ix_iy_i\ge\angle pxy,\quad\liminf_{i\to\infty}\angle p_ix_iz_i\ge\angle pxz.\] Combining these equalities and inequalities yields the desired equality. ◻

Proof. By the definition of angle in Definition 14, it is clear that \(\angle pxy=\angle pxy'\). Thus it remains to show that \(\angle pxy=\angle p'xy\). We may assume that \(p,p',x\) lie on a shortest path in this order. See Figure 4.

Let \(z\in B\) be a point on the extension of the shortest path \(yx\) beyond \(x\). By Lemma 25, we have \[\angle pxy\le\angle p'xy,\quad\angle pxz\le\angle p'xz.\] By Lemma 27, we have \[\angle pxy+\angle pxz=\pi,\quad\angle p'xy+\angle p'xz=\pi.\] Combining these equalities and inequalities yields the desired inequality. ◻

Proof. The proof is a refinement of that of Lemma 27. Let \(z\in B\) be a point on the extension of the shortest path \(yx\) beyond \(x\) such that \(|xy|=|xz|\). Let \(q\in B\) be a point on the extension of the shortest path \(px\) beyond \(x\) such that \(|px|=|qx|\). Set \(s:=|xy|/|px|\) and assume \(s\ll1\). See again Figure 3.

In what follows, we repeat the same argument as in the proof of Lemma 27, but with refined quantitative estimates. We first provide a comparison angle version of Lemma 24. By the Euclidean law of cosine, we have \[\left|\cos\tilde{\angle} pxy-\frac{|px|-|py|}{|xy|}\right|\le s/2\] (since \((a^2+b^2-c^2)/2ab=(a-c)/b+(b^2-(a-c)^2)/2ab\)). The same inequality also holds for \(\tilde{\angle} qxy\). Combining these two inequalities with the triangle inequality \(|py|+|qy|\ge|px|+|qx|\), we obtain \[\label{eq:unif1} \cos\tilde{\angle} pxy+\cos\tilde{\angle} qxy\le s.\tag{2}\] Similarly, we have \[\label{eq:unif2} \cos\tilde{\angle} pxz+\cos\tilde{\angle} qxz\le s.\tag{3}\]

On the other hand, by the explicit version of the almost comparison inequality in Remark 17 and Lemma 19, we obtain \[\label{eq:unif3} \cos\tilde{\angle} pxy+\cos\tilde{\angle} pxz\ge\cos\angle pxy+\cos\angle pxz-\lambda s\ge-\lambda s.\tag{4}\] Similarly, we have \[\label{eq:unif4} \cos\tilde{\angle} qxy+\cos\tilde{\angle} qxz\ge-\lambda s,\tag{5}\] Combining the inequalities 2 , 3 , 4 , 5 yields the desired inequality. ◻

Proof of Proposition 38. The proof is essentially the same as in the GNPC case, [15]. For \(v=(v_1,\dots,v_k)\in\mathbb{R}^k\), we introduce a modified norm \[||v||:=\sum_{i=1}^k\epsilon^i|v_i|,\] where \(\epsilon\) is a positive constant depending only on \(k\), which will be defined below.

By Remark 36, we may assume that \(f\) is a \((k,\delta)\)-strainer map on a neighborhood \(U\) of \(x\). For any \(y\in U\) and \(v\in\mathbb{R}^k\setminus\{f(y)\}\) sufficiently close to \(f(y)\), we show that there exists \(z\in U\) such that \[\label{eq:open} ||f(z)-v||-||f(y)-v||\le-\epsilon'|yz|,\tag{7}\] where \(\epsilon'\) is another positive constant depending only on \(k\). Then by Lemma 41, \(f\) is \(\epsilon'\)-open with respect to \(||\cdot||\). Replacing the modified norm with the Euclidean one, we obtain the claim.

Let us show 7 . Since \(v\neq f(y)\), there is some \(1\le i\le k\) such that \(f_i(y)\neq v_i\). Fix such \(i\). If \(f_i(y)>v_i\) (resp.\(f_i(y)<v_i\)), we choose \(z\) to be a point sufficiently close to \(y\) on the shortest path \(p_iy\) (resp.on the extension of \(p_iy\) beyond \(y\)) such that \(f_i(y)>f_i(z)>v_i\) (resp.\(f_i(y)<f_i(z)<v_i\)). Then we have \[|f_j(z)-v_j|-|f_j(y)-v_j|\le \begin{cases} \delta|yz| & j<i \\ -|yz |& j=i \\ |yz| & j>i \end{cases}.\] Here the first inequality follows from the almost orthogonality 6 , the first variation formula 1 (provided \(z\) is close to \(y\)), and the local differentiability as in Lemma 27 (when \(z\) is beyond \(y\)). The second inequality follows from the choice of \(z\) and the third inequality follows from the \(1\)-Lipschitz property of \(f\).

The above inequalities imply \[\begin{align} ||f(z)-v||-||f(y)-v||&\le\sum_{j=1}^{i-1}\epsilon^j\delta|yz|-\epsilon^i|yz|+\sum_{j=i+1}^k\epsilon^j|yz|\\ &\le-(\epsilon^i/2)|yz|, \end{align}\] provided \(\delta\ll\epsilon\) are small enough compared to \(k\). This completes the proof. ◻

Proof of Lemma 43. By the openness of a strainer, Remark 36, we may assume that \(p_1,\dots,p_k\) is a \((k,\delta)\)-strainer at \(z\). Hence it remains to show that \[|\angle p_izy-\pi/2|<\delta\] for any \(1\le i\le k\). Proposition 33 implies that \(\angle p_izy\) is \((\delta/2)\)-close to \(\tilde{\angle} p_izy\), provided that \(y,z\) are sufficiently close to \(x\) (depending only on the local semiconcavity constant and the length of the strainer, see Definition 35). Furthermore, the assumption that \(||p_iy|-|p_iz||<\delta'|yz|\) implies that \(\tilde{\angle} p_izy\) is \((\delta/2)\)-close to \(\pi/2\), provided \(\delta'\ll\delta\). This completes the proof. ◻

Proof of Proposition 39. Since \(f\) consists of distance functions, it is \(\sqrt k\)-Lipschitz (with respect to the Euclidean norm). By Proposition 38, \(f\) is \(\epsilon_k\)-open. Hence it remains to show that \(f\) is locally injective around \(x\) (see [15] for more details).

Suppose that \(f\) is not locally injective around \(x\). Then there exist \(y_j,z_j\in B\) converging to \(x\) such that \(f(y_j)=f(z_j)\). By the proof of Lemma 43, we see that \[\lim_{j\to\infty}\angle p_iz_jy_j=\pi/2\] for any \(1\le i\le k\). By the extension property of a strainer in Remark 37, we may replace \(y_j\) by \(y_j'\) which has the same property as above and whose distance to \(z_j\) is uniformly bounded below by a positive number. Passing to a subsequence and using the continuity of angle in Lemma 30, we may assume that \(y_j'\) converges to some \(p_{k+1}\neq x\) such that \(\angle p_ixp_{k+1}=\pi/2\) for any \(1\le i\le k\). In particular, \(p_1,\dots,p_k,p_{k+1}\) is a \((k+1,\delta)\)-strainer at \(x\), which contradicts the assumption. ◻

Proof of Lemma 45. We first show the finiteness. Suppose that there exists \(x\in X\) for which the local strainer number is infinite. By definition, for any positive integer \(k\), there exists a \((k,0)\)-strainer \(p_1^k,\dots,p_k^k\) at \(x\). By the extension property of a strainer in Remark 37, we may assume that \(|p_i^kx|\) is uniformly bounded below by a positive number. Using a diagonal argument and the continuity of angle in Lemma 30, one can find an infinite sequence of points \(p_1,p_2,\ldots\in B\setminus\{x\}\) such that \[\angle p_ixp_j=\pi/2\] for all \(i<j\). Passing to a subsequence, we may assume that \(p_i\) converges to \(p_\infty\neq x\). However, taking \(i,j\to\infty\) in the above equality gives \(\angle p_\infty xp_\infty=\pi/2\), which is a contradiction.

Next we show the constancy. Suppose that the local strainer number at \(x\in X\) is \(k\). Then by Proposition 39 and Remark 46, there exists a neighborhood of \(x\) that is homeomorphic to \(\mathbb{R}^k\). Since \(X\) is connected, such \(k\) must be unique. ◻

Proof of Lemma 50. In what follows, we abuse the symbol \(\delta(t)\) to denote various linear functions \(Ct|v|\), whose coefficients \(C>0\) satisfy the dependence described in the statement of Lemma 50.

We first consider the \(s>0\) case (see Figure 5). By definition \[f(c(t))=f(x)+tv,\quad f(c(s))=f(x)+sv,\] which implies \[\label{eq:cur1-1} (t/s)(f(c(s))-f(x))-(f(c(t))-f(x))=0.\tag{8}\] Suppose \(f=(d(p_1,\cdot),\dots,d(p_n,\cdot))\). Since \(c_s(t)\) is on the extension of the shortest path \(xc(s)\), by applying the uniform convergence of angle in Proposition 33 (the explicit version) twice, we have \[\label{eq:cur1-2} \left|\frac{|p_i,c(s)|-|p_ix|}{|x,c(s)|}-\frac{|p_i,c_s(t)|-|p_ix|}{|x,c_s(t)|}\right|<100\lambda\frac{|x,c_s(t)|}{|p_ix|},\tag{9}\] where \(\lambda\ge1\) is the local semiconcavity constant on \(U\). Since \(f\) is \(\epsilon_n\)-bi-Lipschitz by Proposition 39 and \(s>0\), we have \[\label{eq:cur1-3} |x,c_s(t)|=(t/s)|xc(s)|\le\epsilon_n^{-1}t|v|.\tag{10}\] Combining 9 and 10 , we get \[\label{eq:cur1-4} \left|\frac{f(c(s))-f(x)}{|x,c(s)|}-\frac{f(c_s(t))-f(x)}{|x,c_s(t)|}\right|<\delta(t),\tag{11}\] where \(\delta(t)=Ct|v|\) as in the statement of Lemma 50. Combining 10 with 11 again, we have \[\label{eq:cur1-5} \left|(t/s)(f(c(s))-f(x))-(f(c_s(t))-f(x))\right|<\delta(t)t|v|.\tag{12}\] Combining 8 and 12 , we obtain \[|f(c(t))-f(c_s(t))|<\delta(t)t|v|.\] Since \(f\) is \(\epsilon_n\)-bi-Lipschitz, this shows the claim.

Next we consider the \(s<0\) case (see Figure 6). The equality 8 remains true. Instead of 11 , additionally using the local differentiability, Lemma 27, we get \[\left|-\frac{f(c(s))-f(x)}{|x,c(s)|}-\frac{f(c_s(t))-f(x)}{|x,c_s(t)|}\right|<\delta(t).\] This, together with \(|x,c_s(t)|=(-t/s)|x,c(s)|\), implies 12 . The desired inequality now follows from 8 and 12 as before. This completes the proof. ◻

Proof. Let \(0<|s|<t\ll1\). By using the same notation as in Lemma 50, \[\frac{|x,c(s)|}{|s|}=\frac{|x,c_s(t)|}{t}.\] Therefore, by Lemma 50, \[\begin{align} \left|\frac{|x,c(s)|}{|s|}-\frac{|x,c(t)|}{t}\right|&=\left|\frac{|x,c_s(t)|}{t}-\frac{|x,c(t)|}{t}\right|\\ &\le\frac{|c(t),c_s(t)|}{t}<\delta(t)|v|, \end{align}\] which shows the claim. Finally, since a uniform limit of continuous functions is continuous, the value \(|\dot{c}(0)|\) depends continuously on \(x\) and \(v\). ◻

Proof. The first claim is an immediate consequence of Lemmas 50 and 51. Indeed, by the definition of \(\gamma\) and Lemma 51, \(\gamma(|\dot{c}(0)|t)\) is the limit of \(c_{t_i}(t)\), which in turn is \(\delta(t)|t||v|\)-close to \(c(t)\) by Lemma 50.

The second claim follows from the first. Suppose that \(t>0\) and \(g\) is \(L\)-Lipschitz around \(x\). By the first claim, we have \[\label{eq:cur3} \left|\frac{g(c(t))-g(x)}{t}-\frac{g(\gamma(|\dot{c}(0)|t))-g(x)}{t}\right|\le\frac{L|\gamma(|\dot{c}(0)|t),c(t)|}{t}<L\delta(t)|v|.\tag{13}\] Taking \(t\to0\), we obtain \((g\circ c)^+(0)=|\dot{c}(0)|(g\circ\gamma)^+(0)\). The other equality follows in the same way. ◻

Proof. Let \(\gamma(t)\) be a (unit-speed) tangential shortest path for \(c\) as in Lemma 53. Since \(g\) is a distance function, by the uniform convergence of angle in Proposition 33 (the explicit version), we have \[\label{eq:cur4} \left|\frac{g(\gamma(t))-g(x)}{t}-(g\circ\gamma)'(0)\right|<100\tilde{\lambda}\frac{t}{|px|},\tag{14}\] where \(\tilde{\lambda}\ge 1\) is the local semiconcavity constant on \(\tilde{B}\). Combining 14 with 13 and Remark 52 shows the claim. The right-hand side of 14 explains the additional dependency of \(\tilde{\delta}\) on \(\tilde{\lambda}\) and \(|px|\). More precisely, \(\tilde{\delta}\) depends on a lower bound for \(|px|\), and thus it can be chosen uniformly for every \(x'\) close to \(x\). In particular, the convergence \(\lim_{t\to0}(g(c(t))-g(x))/t=(g\circ c)'(0)\) is locally uniform, and hence the limit depends continuously on \(x\) and \(v\). ◻

Proof. Fix \(p\in X\) and set \(g:=d(p,\cdot)\). Let \(f:U\to\mathbb{R}^n\) be a strainer chart on \(X\) that is contained in a normal ball around \(p\). We show that \(g\circ f^{-1}\) is a \(C^{1,1/2}\) function on \(f(U)\setminus\{f(p)\}\). Note that Lemma 54 already showed that it is a \(C^1\) function. More precisely, we observed that the differential of \(g\circ f^{-1}\) is a uniform limit of differential quotients, which are continuous functions. In what follows, we improve it to the local \(1/2\)-Hölder continuity, by examining the previous proof in detail and making use of the explicit error function.

Let \(c_x(t)=f^{-1}(f(x)+tv)\) and \(c_y(t)=f^{-1}(f(y)+tv)\) be strainer curves passing through \(x,y\in U\setminus\{p\}\) in the same direction \(v\in\mathbb{R}^n\), respectively. By Lemma 54, we have \[\label{eq:c1-1} \left|\frac{g(c_x(t))-g(x)}{t}-(g\circ c_x)'(0)\right|<\tilde{\delta}(t)|v|,\quad\left|\frac{g(c_y(t))-g(y)}{t}-(g\circ c_y)'(0))\right|<\tilde{\delta}(t)|v|.\tag{15}\] Note that the error function \(\tilde{\delta}(t)\) can be chosen uniformly for any close \(x\) and \(y\), for the reason described in the proof of Lemma 54. By the \(1\)-Lipschitz property of \(g\) and the \(\epsilon_n\)-bi-Lipschitz property of \(f\) (Proposition 39), we have \[\label{eq:c1-2} \begin{align} \left|\frac{g(c_x(t))-g(x)}{t}-\frac{g(c_y(t))-g(y)}{t}\right|&\le\frac{1}{|t|}\left(|g(c_x(t))-g(c_y(t))|+|g(x)-g(y)|\right)\\ &\le\frac{1}{|t|}(|(c_x(t),c_y(t)|+|xy|)\\ &\le\frac{2\epsilon_n^{-2}}{|t|}|xy|. \end{align}\tag{16}\] Combining 15 and 16 , we obtain \[|(g\circ c_x)'(0)-(g\circ c_y)'(0)|<\frac{2\epsilon_n^{-2}}{|t|}|xy|+2\tilde{\delta}(t)|v|.\] Since \(\tilde{\delta}(t)=\tilde{C}|t||v|\) by Lemma 54, letting \(t:=|xy|^{1/2}\) shows the claim. ◻

Proof of Lemma 59. The proof is the same as the original one, [20], using the local differentiability proved in Lemma 27.

Let \(p_1,\dots,p_n\in B\) be the \((n,\delta_n')\)-strainer that defines \(f\), where \(B\) is a normal ball and \(\delta_n'\) is the constant of Proposition 38. Fixing \(x\in U\), we consider an opposite strainer \(q_1,\dots,q_n\in B\), that is, \(q_i\) is a point on the extension of a shortest path \(p_ix\) beyond \(x\). By the local differentiability, Lemma 27 and Remark 28, \(q_1,\dots,q_n\) is indeed an \((n,\delta_n')\)-strainer at \(x\). We show that for a sufficiently small neighborhood \(V\subset U\) of \(x\), \[g:=\frac{1}{n}\sum_{i=1}^n g_i,\quad g_i:=d(q_i,\cdot)\] is the desired special function on \(V\) (clearly \(g\) is \(1\)-Lipschitz).

Let \(\gamma\) be a shortest path such that \(\gamma(0)\in V\). Suppose \((f_i\circ\gamma)^+(0)\le \delta\) for all \(1\le i\le n\), where \(\delta=\delta_n\) will be determined later. Then we have \[\label{eq:dc1-1} (g_i\circ\gamma)^+(0)\ge-2\delta,\tag{18}\] provided \(V\) is sufficiently small (indeed, by the local differentiability, Lemma 27 and Remark 28, and the continuity of angle, Lemma 30, the sum of \((f_i\circ\gamma)^+(0)\) and \((g_i\circ\gamma)^+(0)\) must be almost zero). On the other hand, by Proposition 39, the strainer map \((g_1,\dots,g_n)\) is \(\epsilon\)-bi-Lipschitz on \(V\) for some \(\epsilon\) depending only on \(n\). This implies that \(\sum_{i=1}^n|(g_i\circ\gamma)^+(0)|\ge\epsilon\), and hence there exists \(1\le i_0\le n\) such that \[\label{eq:dc1-2} |(g_{i_0}\circ\gamma)^+(0)|\ge\epsilon/n.\tag{19}\] Choosing \(\delta\ll\epsilon\) and combining 18 and 19 , we obtain the desired property 17 for some suitable \(\epsilon_n\) and \(\delta_n\). ◻

Proof of Lemma 60. The proof is almost identical to the original one, [18], with minor modifications mainly based on Lemma 53.

Let \(f\) and \(g\) be as in the assumption of Lemma 60. Let \(c(t):=f^{-1}(f(x)+tv)\) be a strainer curve with \(|v|=1\), as in Definition 49. To prove that \(g\circ f^{-1}\) is semiconcave, it suffices to show that \[\begin{gather} (g\circ c)^+(0)+(g\circ c)^-(0)\le 0,\tag{20}\\ (g\circ c)(t)\le (g\circ c)(0)+(g\circ c)^+(0)t+Ct^2,\tag{21} \end{gather}\] for any small \(t>0\), where \(C\) is a uniform positive constant independent of \(x,t,v\). The other inequality for \(t<0\) and \((g\circ c)^{-}(0)\) similar to 21 follows by replacing \(v\) with \(-v\). Note that our notation \((g\circ c)^\pm(0)\) represents the directional derivative of \(g\) in the direction \(c^\pm(0)\) rather than the right/left derivative.

Let \(\gamma (t)\) be a (unit-speed) tangential shortest path for \(c\) as in Lemma 53. Then the first inequality 20 follows from the second statement of Lemma 53, since \(g\) is semiconcave and hence 20 with \(c\) replaced by \(\gamma\) holds (notice that this part is different from the original proof in [18]).

Let us prove the second inequality 21 in the same way as the original proof. In what follows we denote by \(C\) various (large) positive constants depending only on \(n\), the local semiconcavity constant on \(U\), the strainer length of \(f\), and the semiconcavity of \(g\). Let \(f_i\) denote the \(i\)-th component of \(f\), where \(1\le i\le n\). By the semiconcavity of \(f_i\) and \(g\) and Remark 52, we have \[\begin{gather} (f_i\circ\gamma)(|\dot{c}(0)|t)\le f_i(x)+|\dot{c}(0)|(f_i\circ\gamma)^+(0)t+Ct^2,\tag{22}\\ (g\circ\gamma)(|\dot{c}(0)|t)\le g(x)+|\dot{c}(0)|(g\circ\gamma)^+(0)t+C t^2.\tag{23} \end{gather}\] By definition, \(f_i\circ c(t)=f_i(x)+tv_i\). This, together with Lemma 53 and 22 , implies that \[(f_i\circ\gamma)(|\dot{c}(0)|t)\le(f_i\circ c)(t)+Ct^2.\] By the \(\epsilon_n\)-openness of \(f\), Proposition 38, we can find \(y\in U\) such that \[\label{eq:dc2-5} f_i(y)\le(f_i\circ c)(t),\quad|y,\gamma(|\dot{c}(0)|t)|\le Ct^2\tag{24}\] for all \(1\le i\le n\) (more specifically, we choose \(y\) near \(\gamma(|\dot{c}(0)|t)\) such that \(f_i(y)=\min\{(f_i\circ c)(t),(f_i\circ\gamma)(|\dot{c}(0)|t)\}\)). Combining the second inequality of 24 with 23 by the \(1\)-Lipschitz property of \(g\), we have \[\label{eq:dc2-6} g(y)\le g(x)+(g\circ c)^+(0)t+Ct^2,\tag{25}\] where we also used Lemma 53. Moreover, as we will see below, the first inequality of 24 implies \[\label{eq:dc2-7} (g\circ c)(t)\le g(y).\tag{26}\] Combining 25 and 26 proves 21 .

It remains to prove 26 . For fixed \(t\), consider a nonempty compact subset \[A:=\left\{z\in U\;\middle|\; \begin{gather} f_i(y)\le f_i(z)\le (f_i\circ c)(t)\\ (g\circ c)(t)\le g(z) \end{gather} \right\},\] where \(i\) runs over all integers from \(1\) to \(n\). The nonemptiness of \(A\) follows from the first inequality of 24 , taking \(z=c(t)\); the compactness follows from the first inequalities for \(f_i\) and the fact that \(f\) is a homeomorphism. Since \(f\) is injective, to prove 26 , it suffices to show that \(f(y)\in f(A)\). Suppose otherwise. Since \(A\) is compact, one finds \(z\in A\) such that \(f(z)\neq f(y)\) is closest to \(f(y)\).

Set \(\tilde{v}:=f(y)-f(z)\) and consider a new strainer curve \(\tilde{c}(s):=f^{-1}(f(z)+s\tilde{v})\) passing through \(z\). Clearly, for sufficiently small \(s>0\), we have \[\label{eq:dc2-8} f_i(y)\le (f_i\circ\tilde{c})(s)\le f_i(z)\tag{27}\] for all \(1\le i\le n\). In particular, \((f_i\circ\tilde{c})^+(0)\le 0\). Let \(\tilde{\gamma}\) be a tangential shortest path for \(\tilde{c}\) as in Lemma 53. By the second statement of Lemma 53, we have \[(f_i\circ\tilde{\gamma})^+(0)\le 0.\] By the assumption that \(g\) is an \((\epsilon,\delta)\)-special function, 17 , we get \((g\circ\tilde{\gamma})^+(0)\ge\epsilon\). By Lemma 53 again, this implies \((g\circ\tilde{c})^+(0)>0\). Thus, for sufficiently small \(s>0\), we have \[\label{eq:dc2-9} (g\circ\tilde{c})(s)>g(z).\tag{28}\] The inequalities 27 and 28 , together with the fact that \(z\in A\), imply \(\tilde{c}(s)\in A\). However, \((f\circ\tilde{c})(s)\) is closer to \(f(y)\) than \(f(z)\), which is a contradiction. ◻

Proof of Proposition 58. The proof is the same as in [18], [20]. Let \(f:U\to\mathbb{R}^n\) be a strainer chart on \(X\). By the local semiconcavity, \(f\) is a DC map. It remains to show that \(f^{-1}\) is a DC map. We show that if \(h\) is a semiconcave function on \(U\), then \(h\circ f^{-1}\) is a DC function. For any fixed \(x\in U\), let \(g\) be an \((\epsilon_n,\delta_n)\)-special function around \(x\) as in Lemma 59. Consider the decomposition \[h\circ f^{-1}=(h\circ f^{-1}+Lg\circ f^{-1})-Lg\circ f^{-1},\] where \(L\gg1\). By Lemma 60, the second term \(Lg\circ f^{-1}\) is a semiconcave function. Furthermore, if \(L\) is large enough (compared to the local Lipschitz constant of \(h\)), then \((h+Lg)\) is also a \((1,\delta_n)\)-special function for \(f\). Therefore, by Lemma 60, the first term \((h+Lg)\circ f^{-1}\) is also semiconcave, as desired. ◻

Proof. Since the continuity of \(|\dot{c}(0)|\) was already proved in Lemma 51, it suffices to check the well-definedness of \(F\).

Let \(f:U\to\mathbb{R}^n\) be a strainer chart around \(x\in X\) and \(c(t)=f^{-1}(f(x)+tv)\) a strainer curve in the direction \(v\in\mathbb{R}^n\). Let \(g:V\to\mathbb{R}^n\) be another strainer chart around \(x\). Set \[\tilde{c}(t):=g^{-1}(g(x)+t\tilde{v}),\quad\tilde{v}:=d(g\circ f^{-1})(v)=(g\circ c)'(0).\] We show that \[\label{eq:fin1} |c(t),\tilde{c}(t)|<\delta(t)|t|,\tag{29}\] where \(\delta\) is a positive function such that \(\delta(t)\to0\) as \(t\to0\). Then, by the triangle inequality, we have \(|\dot{c}(0)|=|\dot{\tilde{c}}(0)|\), as desired. Since \(g\) is bi-Lipschitz, 29 is equivalent to \(|g\circ c(t), g(x)+t\tilde{v}|<\delta(t)|t|\). The latter immediately follows from the \(C^1\) smoothness of \(g\circ f^{-1}\) proved in Proposition 56. ◻

Proof. The proof is similar to that of Proposition 56. Let \(f:U\to\mathbb{R}^n\) be a strainer chart on \(X\). Let \(c_x(t)=f^{-1}(f(x)+tv)\) and \(c_y(t)=f^{-1}(f(y)+tv)\) be strainer curves passing through \(x,y\in U\) in the same direction \(v\in\mathbb{R}^n\), respectively. By Lemma 51, we have \[\label{eq:fin2-1} \left||\dot{c}_x(0)|-\frac{|x,c_x(t)|}{|t|}\right|<\delta(t)|v|,\quad\left||\dot{c}_y(0)|-\frac{|y,c_y(t)|}{|t|}\right|<\delta(t)|v|,\tag{30}\] By the \(\epsilon_n\)-bi-Lipschitz property of \(f\) (Proposition 39), we have \[\label{eq:fin2-2} \left|\frac{|x,c_x(t)|}{|t|}-\frac{|y,c_y(t)|}{|t|}\right|\le\frac{1}{|t|}{(|xy|+|c_x(t),c_y(t)|)}\le\frac{2\epsilon_n^{-2}}{|t|}|xy|.\tag{31}\] Combining 30 and 31 , we obtain \[\left||\dot{c}_x(0)|-|\dot{c}_y(0)|\right|<\frac{2\epsilon_n^{-2}}{|t|}|xy|+2\delta(t)|v|.\] Since \(\delta(t)=C|t||v|\) by Lemma 51, letting \(t:=|xy|^{1/2}\) shows the claim. ◻

Proof. In what follows, we fix \(x\) and consider the restriction of \(F\) to \(T_xX\). Since a strainer chart is bi-Lipschitz, \(F\) is positive definite. By definition, \(F\) is absolutely homogeneous. It remains to prove the triangle inequality. Let \(v,w\in T_xX\). For small \(t>0\), set \[\begin{align} c(t)&:=f^{-1}(f(x)+tv),\\ \tilde{c}(t)&:=f^{-1}(f(c(t))+tw)=f^{-1}(f(x)+t(v+w)). \end{align}\] See Figure 7.

By the triangle inequality for the original distance, we have \[|x,\tilde{c}(t)|\le|x,c(t)|+|c(t),\tilde{c}(t)|.\] Dividing both sides by \(t\) and taking \(t\to0\), we get \[\label{eq:fin3-1} F(x,v+w)\le F(x,v)+\lim_{t\to0}\frac{|c(t),\tilde{c}(t)|}{t}.\tag{32}\] Furthermore, by the uniform convergence of norm in Lemma 51, we see that \[\left|\frac{|c(t),\tilde{c}(t)|}{t}-F(c(t),w)\right|<\delta(t)|w|.\] Taking \(t\to0\) and using the continuity of \(F\), we obtain \[\label{eq:fin3-2} \lim_{t\to0}\frac{|c(t),\tilde{c}(t)|}{t}=F(x,w).\tag{33}\] Combining 32 and 33 completes the proof. ◻

Proof. Since \(d\) is an intrinsic distance, it is written as \[d(p,q)=\inf_{c}L(c),\quad L(c):=\sup\sum_{i=1}^Nd(c(t_{i-1}),c(t_i)),\] where \(c:[0,1]\to X\) runs over all (not necessarily \(C^1\)) curves between \(p\) and \(q\), and the supremum is taken over all finite subdivisions \(0=t_0<t_1<\dots<t_N=1\). Since any shortest path in \(X\) is a \(C^1\) curve by Proposition 56, in order to prove that \(d=\tilde{d}\), it suffices to show that \(L(c)=\tilde{L}(c)\) for any \(C^1\) curve \(c\).

Let \(c:[0,1]\to X\) be a \(C^1\) curve. Take a sufficiently fine subdivision \(0=t_0<t_1<\dots<t_N=1\) and set \(\Delta:=\max_{1\le i\le N}(t_i-t_{i-1})\ll1\). By the uniform convergence of norm in Lemma 51 and the \(C^1\) smoothness of \(c\), we have \[\left||c(t_{i-1}),c(t_i)|-F(c(t_{i-1}),c'(t_{i-1}))(t_i-t_{i-1})\right|<M\delta(\Delta)(t_i-t_{i-1}),\] where \(M\) is a positive number depending on the maximum speed of \(c\). By adding all the inequalities and taking \(\Delta\to0\), we obtain \(L(c)=\tilde{L}(c)\). ◻

Proof. This follows from the fact that every space of directions of \(X\) is a unit sphere (see [11], [20]). Indeed, let \(p,x,y,z\in X\) be points in a sufficiently small ball such that \(y,x,z\) lie on a single shortest path. Suppose that the local differentiability does not hold for these points. Together with Lemma 22, this implies \[\angle pxy+\angle pxz>\pi.\] Consider the space of directions \(\Sigma_xX\) at \(x\) and denote by \(p',y',z'\in\Sigma_xX\) the elements defined by the shortest paths \(xp, xy, xz\), respectively. Since our angle is compatible with the distance on the space of directions in the CAT setting, the above inequality implies \(|p',y'|+|p',z'|>\pi\). Since \(y'\) and \(z'\) are antipodal points in \(\Sigma_xX\), this contradicts the fact that \(\Sigma_xX\) is a unit sphere. ◻

Proof. Choose \(l>0\) such that the closed \(l\)-neighborhood of \(K\) is contained in \(B\). Extend the shortest path \(xy\) beyond \(y\) and let \(y'\in B\) be a new endpoint such that \(|xy'|=l\). See Figure 8.

By Lemma 22, we have \[\label{eq:unif391} \angle pyx+\angle pyy'\ge \pi.\tag{34}\] Due to local differentiability, the angle varies continuously (cf.Lemma 30). Therefore, by a compactness argument using the Arzelà–Ascoli theorem [28], we get the following: for any positive number \(\delta\), if the distance between \(x\) and \(y\) is small enough, then \[\label{eq:unif392} |\angle pxy-\angle pyy'|<\delta.\tag{35}\] Indeed, since \(\angle pxy=\angle pxy'\), the above two angles converge to the same angle when \(x\) and \(y\) converge to the same point \(z\in K\) and \(xy'\) converges to some shortest path emanating from \(z\). Hence 35 is proved by contradiction. Note that here we used the assumptions that \(x,y\in K\) and \(|xy'|=l\) to ensure that the limit triangle does not degenerate at the limit point \(z\). Furthermore, by the almost comparison inequality ?? , we have \[\label{eq:unif393} \angle pxy+\angle pyx\le\tilde{\angle} pxy+\tilde{\angle} pyx+\delta\le\pi+\delta,\tag{36}\] provided that \(|xy|/|pK|\) is small enough compared to \(\delta\). Combining the inequalities 34 , 35 , and 36 , we obtain the desired inequality. ◻

Alternative proof of Proposition 70. As before, choose \(l>0\) such that the closed \(l\)-neighborhood of \(K\) is contained in \(B\). Extend the shortest path \(px\) beyond \(x\) and let \(q\in B\) be a new endpoint such that \(|qx|=l\). See Figure 9.

By Lemma 24, we have \[\label{eq:unif39391} \angle pxy+\angle qxy\ge\pi\tag{37}\] (in fact the equality holds, due to local differentiability and Remark 28). On the other hand, for any positive number \(\delta\), if \(x\) and \(y\) are sufficiently close, then \[\label{eq:unif39392} \angle pyx+\angle qyx\ge\pi-\delta.\tag{38}\] This immediately follows from 35 and Lemma 24. Indeed, take a point \(x'\in B\) on the extension of the shortest path \(yx\) beyond \(x\) with \(|x'y|=l\). Then by the same argument as 35 , the two angles in 38 are \(\delta/2\)-close to \(\angle pxx'\) and \(\angle qxx'\), respectively, provided \(|xy|\) is small enough. The sum of the latter two angles is not less than \(\pi\) by Lemma 24, which gives 38 . Furthermore, by the almost comparison inequality ?? as before, we have \[\begin{gather} \angle pxy+\angle pyx\le\tilde{\angle} pxy+\tilde{\angle} pyx+\delta\le\pi+\delta,\tag{39}\\ \angle qxy+\angle qyx\le\tilde{\angle} qxy+\tilde{\angle} qyx+\delta\le\pi+\delta,\tag{40} \end{gather}\] provided \(|xy|\) is small enough compared to \(|pK|\) and \(l\). Combining 37 , 38 , 39 , and 40 , we obtain the desired inequality. Note that the difference from the previous proof is that we used Lemma 24 here instead of Lemma 22. ◻

Proof of Theorem 2. In what follows, we will consider the statements in Sections 4.2, 4.3, 4.4, and 4.5 with local semiconcavity replaced by local semiconvexity plus local differentiability, as already done for Section 4.1.

Regarding Section 4.2, the definition and properties of a strainer are exactly the same as before, since it only involves the first-order derivative of the distance function, i.e., angles. Note that the strainer argument takes place in a small neighborhood of a fixed strained point (compared to the length of a strainer), so the additional dependency of the error function \(\delta\) of Proposition 70 does not matter.

Regarding Section 4.3, the properties of a strainer curve remain true with the same proofs, except for the explicit representation of the error function \(\delta\). Every statement holds for an implicit error function arising from Proposition 70.

Regarding Section 4.4, due to the lack of the explicit error function, we only get a \(C^1\) structure in Proposition 56, which immediately follows from Lemma 54. On the other hand, Proposition 58 on the DC structure remains true, but requires slight modifications, as it involves the second-order derivative of the distance function. In fact, it suffices to change the sign in the following manner: a general semiconvex function \(f\) is defined by the condition that \(-f\) is semiconcave; a DC function here should be defined as the difference of two semiconvex functions, which in turn is equivalent to the previous definition; then all the previous arguments carry over to the current setting simply by replacing a function \(f\) with \(-f\). For example, the definition of an \((\epsilon,\delta)\)-special function in 17 will be replaced by \[(f_i\circ\gamma)^+(0)\ge-\delta\text{ for all }1\le i\le n\implies (g\circ\gamma)^+(0)\le-\epsilon.\] Then Lemmas 59 and 60 are valid for locally semiconvex, locally differentiable G-spaces, by the same proofs with the sign change explained above. The details are left to the reader.

Regarding Section 4.5, we can repeat exactly the same construction of a Finsler metric, except for Lemma 65 on the \(1/2\)-Hölder continuity. This completes the proof of Theorem 2. ◻

Proof. Assume that \(y\in B\) is sufficiently close to \(x\). By the upper semicontinuity of angle, Lemma 21, we have \[\angle p_iyp_j<\pi/2+\delta,\quad \angle p_iyq_j'<\pi/2+\delta,\] where \(q_j'\in B\) is a point on the extension of the shortest path \(p_jy\) beyond \(y\). On the other hand, by Lemma 22, the sum of these two angles is not less than \(\pi\). Therefore, we also have \(\angle p_iyp_j>\pi/2-\delta\) and \(\angle p_iyq_j'>\pi/2-\delta\), as required. ◻

Proof of Lemma 79. By the openness of the strainer condition, Lemma 76, we may assume that \(p_1,\dots,p_k\) is a \((k,\delta)\)-strainer at \(y\). Hence it remains to show that \[\label{eq:bilip391} |\angle p_iyx-\pi/2|<\delta,\quad|\angle p_iyy'-\pi/2|<\delta,\tag{42}\] where \(y'\) is a point on the extension of the shortest path \(xy\) beyond \(y\).

In what follows, we repeat the same argument as in the proof of Proposition 70, replacing the continuity of angle (cf.Lemma 30) with the upper semicontinuity of angle, Lemma 21. See again Figure 8, regarding \(p\) as \(p_i\) and ignoring \(K\).

By Lemma 22 on adjacent angles, we have \[\angle p_iyx+\angle p_iyy'\ge \pi.\] Since \(y\) is sufficiently close to \(x\), the upper semicontinuity of angle in Lemma 21 implies \[\angle p_iyy'\le \angle p_ixy+\delta',\] where \(\delta'\) is as in the statement of Lemma 79 (compare the above inequality with 35 where \(x\) is not fixed). By the almost comparison inequality ?? , we have \[\angle p_ixy+\angle p_iyx\le\tilde{\angle} p_ixy+\tilde{\angle} p_iyx+\delta'\le\pi+\delta',\] provided that \(|xy|/|p_ix|\) is small enough compared to \(\delta'\). Combining the above three inequalities, we obtain \[\label{eq:bilip392} |\angle p_iyx-\tilde{\angle} p_iyx|<2\delta',\quad|\angle p_i yx+\angle p_iyy'-\pi|<2\delta'.\tag{43}\] Now, using the assumption that \(||p_ix|-|p_iy||<\delta'|xy|\), we have \(|\tilde{\angle} p_iyx-\pi/2|<2\delta'\), provided \(\delta'\) is small enough. Together with 43 , this implies the desired inequality 42 , provided \(\delta'\ll\delta\). ◻

Proof. The proof is along the same lines as that of [20]. Let \(X\) be a locally semiconvex G-space and \(B\) a normal ball of radius \(r>0\). Let us denote by \(10\bar B\) the concentric closed ball of radius \(10r\).

Let \(\delta>0\) be small enough, which will be determined later. Take a finite \(\delta\)-net \(\{p_\alpha\}_{\alpha=1}^N\) of \(10\bar B\), i.e., the closed \(\delta\)-balls around \(p_\alpha\) cover \(10\bar B\). We prove that the distance map \[f:=(d(p_1,\cdot),\dots,d(p_N,\cdot)):B\to\mathbb{R}^N\] is a locally \((1-\varkappa(\delta))\)-bi-Lipschitz embedding with respect to the sup-norm of \(\mathbb{R}^N\), where \(\varkappa(\delta)\) is a positive function such that \(\varkappa(\delta)\to0\) as \(\delta\to 0\). Note that \(f\) is clearly \(1\)-Lipschitz.

In what follows, we abuse \(\varkappa(\delta)\) to denote various functions satisfying the above property. Let \(x,y\in B\) be such that \(|xy|\le\delta\). Extend the shortest path \(yx\) beyond \(x\) and let \(z,w\in 10\bar B\) be points on the extension such that \(|xz|=r'\), \(|xw|=9r\), respectively, where \(\delta\ll r'\ll r\) will be determined later. Since \(\{p_\alpha\}_{\alpha=1}^N\) is a \(\delta\)-net, there exists \(1\le \alpha\le N\) such that \(p_\alpha\) is \(\delta\)-close to \(w\). See Figure 10.

Since \(\tilde{\angle} wxz=0\), we have \[\tilde{\angle} p_\alpha xz<\varkappa(\delta),\] provided that \(\delta\) is sufficiently small compared to \(r'\). On the other hand, the almost comparison inequality ?? and Lemma 22 on adjacent angles imply \[\tilde{\angle} p_\alpha xz+\tilde{\angle} p_\alpha xy\ge\angle p_\alpha xz+\angle p_\alpha xy-\varkappa(\delta)\ge\pi-\varkappa(\delta),\] provided that \(r'/r\) is sufficiently small compared to the local semiconvexity constant on \(10\bar B\). Combining the above two inequalities, we obtain \[\tilde{\angle} p_\alpha xy\ge\pi-\varkappa(\delta).\] Since \(|xy|\le\delta\ll r\), this implies \[|p_\alpha y|-|p_\alpha x|\ge (1-\varkappa(\delta))|xy|.\] Therefore, \(f\) is locally \((1-\varkappa(\delta))\)-bi-Lipschitz with respect to the sup-norm.

Finally, we modify \(f\) to a global \((1-\varkappa(\delta))\)-bi-Lipschitz embedding. Take a finite \(\delta'\)-net \(\{q_\beta\}_{\beta=1}^M\) in \(\bar B\), where \(0<\delta'\ll\delta\), and consider the distance map \[g:=(d(q_1,\cdot),\dots,d(q_M,\cdot)):B\to\mathbb{R}^M.\] Then the triangle inequality shows that \(g\) satisfies the \((1-\varkappa(\delta'/\delta))\)-bi-Lipschitz property for any \(x,y\in B\) with \(|xy|\ge\delta\). Therefore the map \((f,g):B\to\mathbb{R}^{N+M}\) gives a global almost isometric embedding. This completes the proof. ◻

Proof of Theorem 5. Let \(X\) be a locally semiconvex G-space and \(B\) a normal ball. By the topological homogeneity of \(X\), it suffices to find a manifold point in \(B\).

By Lemma 80, \(B\) has finite Hausdorff dimension, say at most \(N\). Let \[\delta:=\min_{1\le k\le N+1}\delta_k,\] where \(\delta_k\) is a constant of Proposition 78 depending only on \(k\). By Proposition 78, if \(B\) contains a \((k,\delta)\)-strained point, where \(1\le k\le N+1\), then the Hausdorff dimension of \(B\) is at least \(k\). Hence there exists a maximal strained point \(x\) in \(B\), that is, \(x\) is an \((n,\delta)\)-strained point and there are no \((n+1,\delta)\)-strained points in \(B\), where \(1\le n\le N\).

Let \(f:U\to\mathbb{R}^n\) be an \((n,\delta)\)-strainer map around \(x\). By Lemma 79 and the above choice of \(n\), we have for any \(y\in U\), \[\label{eq:liminf} \liminf_{z\to y}\frac{|f(y)-f(z)|}{|yz|}>\delta',\tag{44}\] where \(\delta'\) is a positive constant depending only on \(\delta\).

Now, applying [51], we see that there exists an open (dense) subset of \(U\) on which \(f\) is a locally bi-Lipschitz embedding. Since \(f\) is open by Proposition 78, this gives a manifold point. For the convenience of the reader, we recall an outline of [51] below.

For any positive integer \(m\), we define a closed subset \(U_m\) of \(U\) by \[U_m:=\{y\in U\mid |yz|<1/m,\;z\in U \implies|f(y)-f(z)|\ge\delta'|yz|.\}\] By the inequality 44 , \(\{U_m\}_m\) covers \(U\). Clearly, \(f\) is locally bi-Lipschitz on each \(U_m\). Consider a countable covering of \(U\) by locally closed subsets \(V_m:=U_m\setminus U_{m-1}\). Then the Baire category theorem implies that the union of the interior of \(V_m\) is dense in \(U\) (see [51]). This completes the proof. ◻