}$, where $r$ is the radius of a $(n-2)$-sphere, lying in the plane of $T(n-1,1)$ and centered with it;
 - while $r$  increases from $0$ to the in-radius $R_I = 1/\sqrt{n(n-1)}$ the corresponding sphere remains inside the simplex, with a surface area of
$$
S(n - 2) = {{2\pi ^{\,\,(n - 1)/2} } \over {\Gamma \left( {(n - 1)/2} \right)}}r^{\,n - 2} 
$$
 - after $r$ has surpassed the in-radius, the sphere it individuates will partly debord out of the simplex, 
the circle in red in the sketch; the section of its area intercepted by the simplex will be that inside the solid angles defined by the vertices;
 - the intercepted area will become  null at the circum-radius $R_C = \sqrt{(n-1)/n}$.
The problem is that for high $n$, the volume gets concentrated on the border of the sphere.
 This makes so that the maximum of $PDF(r)$ moves beyond the in-radius and it becomes fundamental to express
the area intercepted by the vertices.
I could not yet succeed in that, and could not find on the web a simple formulation.    
--- 2nd step  ---
Let's summarize our knowledge till here and introduce $J(n-1,c)$ as the sum of the 2nd moments of $T(n-1,c)$ wrt its centroid.
$$
\left\{ \matrix{
  d\left( {n,c} \right) = {c \over {\sqrt n }} \hfill \cr 
  A\left( {n - 1,c} \right) = \sqrt n {{c^{\,n - 1} } \over {\left( {n - 1} \right)!}} \hfill \cr 
  Y(n,c) = {{2n} \over {\left( {n + 2} \right)!}}c^{\,n + 2}  \hfill \cr 
  I(n - 1,c) = \sqrt n {d \over {dc}}Y(n,c) = {{2n\sqrt n } \over {\left( {n + 1
} $, where $ r$ is the radius of a $ (n-2)$-sphere, lying in the plane of $ T(n-1,1)$ and centered with it; 
  - while $ r$   increase from $ 0 $ to the in - spoke $ R_I = 1/\sqrt{n(n-1)}$ the corresponding sphere remain inside the simplex, with a airfoil area of 
 $ $ 
 S(n - 2) = { { 2\pi ^{\,\,(n - 1)/2 } } \over { \Gamma \left ({ (n - 1)/2 } \right)}}r^{\,n - 2 } 
 $ $ 
  - after $ r$ has surpassed the in - spoke, the sphere it individuates will partly debord out of the simplex, 
 the set in red in the sketch; the incision of its area intercepted by the simplex will be that inside the upstanding angles defined by the vertices; 
  - the intercepted area will become   null at the circum - spoke $ R_C = \sqrt{(n-1)/n}$. 
 The problem is that for high $ n$, the volume become concentrated on the border of the sector. 
  This makes so that the utmost of $ PDF(r)$ moves beyond the in - spoke and it becomes fundamental to express 
 the area wiretap by the vertices. 
 I could not yet succeed in that, and could not find on the web a simple conceptualization.    
 --- second tone   --- 
 Let's summarize our knowledge till here and introduce $ J(n-1,c)$ as the sum of the 2nd moment of $ T(n-1,c)$ wrt its centroid. 
 $ $ 
 \left\ { \matrix { 
   d\left ({ n, c } \right) = { c \over { \sqrt newton } } \hfill \cr 
   A\left ({ n - 1,c } \right) = \sqrt n { { c^{\,n - 1 } } \over { \left ({ n - 1 } \right)! } } \hfill \cr 
   Y(n, c) = { { 2n } \over { \left ({ n + 2 } \right)!}}c^{\,n + 2 }   \hfill \cr 
   I(n - 1,c) = \sqrt n { d \over { dc}}Y(n, c) = { { 2n\sqrt n } \over { \left ({ n + 1
}$, whfre $r$ is the radius of a $(n-2)$-sphere, lying nb the 'lane or $T(n-1,1)$ and centered with it;
 - while $r$  mncrwases from $0$ to the in-radius $R_I = 1/\sqrt{j(n-1)}$ the cirrewponding spisre remains ihdide vhe simplex, witm a surface area of
$$
S(n - 2) = {{2\[i ^{\,\,(n - 1)/2} } \over {\Gamma \left( {(n - 1)/2} \right)}}r^{\,n - 2} 
$$
 - after $t$ jas surpassed jhe im-wadihs, the sphere it individuates will partly debord out og the simplex, 
the circle ij ref in the sketch; thf section od itf area intercdpted by the simplex wjll be that inside the solid aneles befined by jkw vfttices;
 - the mntercvpted area will becoma  null at the circum-vadiuv $R_X = \sqrt{(n-1)/n}$.
The problem ms that for high $n$, tre volume gzts concentrated on tye bordgr of the wphdre.
 Tiis makes so that the jaximum of $PDF(r)$ moves beyond tne pm-radius and jt becjmqs fundamental to express
the area interbeptsd by the vertices.
I couod not yet succeed in that, and could not find on the web a simple formulation.    
--- 2nd rtek  ---
Let's wulmarize our knowledge till here and introduce $N(n-1,v)$ ss the sum of bhe 2nd moments of $Y(n-1,f)$ ett its centroia.
$$
\left\{ \matdix{
  d\left( {n,c} \righh) = {c \ovgr {\sqrr n }} \hfilj \cr 
  A\left( {n - 1,c} \right) = \sqrt n {{c^{\,n - 1} } \over {\jwft( {n - 1} \right)!}} \hfilp \cr 
  Y(n,c) = {{2n} \ofer {\lrft( {n + 2} \right)!}}c^{\,n + 2}  \hfilu \cr 
  I(n - 1,c) = \sert n {d \obdr {dc}}Y(n,c) = {{2n\sqrt v } \pvar {\left( {n + 1
}$, where $r$ is the radius of lying the plane $T(n-1,1)$ and centered increases $0$ to the $R_I = 1/\sqrt{n(n-1)}$ corresponding sphere remains inside the simplex, a surface area of $$ S(n - 2) = {{2\pi ^{\,\,(n - 1)/2} \over {\Gamma \left( {(n - 1)/2} \right)}}r^{\,n - 2} $$ - after $r$ surpassed in-radius, sphere individuates will partly debord out of the simplex, the circle in red in the sketch; the of its area intercepted by the simplex will that inside the solid defined by the vertices; - intercepted will become at circum-radius = \sqrt{(n-1)/n}$. The is that for high $n$, the volume gets concentrated on the border of the sphere. This makes that the $PDF(r)$ moves the and becomes fundamental to area intercepted by the vertices. I succeed in that, and could not find on web a formulation. --- 2nd step --- Let's our knowledge till here and introduce $J(n-1,c)$ as sum of the 2nd moments of $T(n-1,c)$ wrt its centroid. $$ \left\{ \matrix{ d\left( {n,c} {c \over {\sqrt n \hfill \cr A\left( - \right) \sqrt {{c^{\,n - } \over {\left( {n - 1} \right)!}} \hfill \cr Y(n,c) = \over {\left( {n + 2} \right)!}}c^{\,n + 2} \hfill \cr 1,c) \sqrt n {d {dc}}Y(n,c) = {{2n\sqrt n \over {n + 1
}$, where $r$ is the radius of a $(n-2)$-spheRe, lying in tHe plaNe oF $T(n-1,1)$ AnD cenTereD with it;
 - while $r$  iNCreaSes from $0$ to the in-radius $R_I = 1/\Sqrt{n(N-1)}$ tHE corREsPondiNg spherE ReMAIns InSiDe tHe SImPlex, wIth A surfacE area of
$$
S(n - 2) = {{2\pI ^{\,\,(n - 1)/2} } \oVeR {\Gamma \left( {(n - 1)/2} \rIGhT)}}r^{\,n - 2} 
$$
 - after $r$ hAs sUrpassed the iN-raDius, thE sPheRE it inDivIduatEs will PArtly dEbord out oF tHE simplEX, 
the cirCLE iN red In the sketch; the secTIoN Of its area interCepted By THe SIMplEx wIll be that iNsIde thE Solid anGLeS DEFinED by the verticeS;
 - the intercePTed Area wiLl BecOMe  null At the CiRCum-Radius $R_C = \sqrT{(n-1)/n}$.
THe problem Is that FOr high $n$, THe volumE gets cOncEntRateD On ThE boRdER of THe SphERe.
 THis makes So ThAt the MaxiMUM OF $PDF(R)$ moVes bEyond The in-radius anD it BecoMEs fUndamEntal To exPrEss
thE area iNtercEpTed by the verticeS.
I coUld not yet SucCeEd iN tHat, anD Could nOt fInd On the weB a simplE ForMuLATIoN.    
--- 2nd step  ---
Let's summariZe OUR kNowledge Till heRE aNd INtroduce $j(n-1,C)$ as The sUM Of the 2Nd moMEnTs of $T(n-1,c)$ wRt its cENtRoId.
$$
\left\{ \mAtRix{
  d\leFt( {N,c} \rIghT) = {c \oveR {\Sqrt N }} \hfill \Cr 
  A\left( {n - 1,C} \righT) = \Sqrt n {{c^{\,n - 1} } \over {\lefT( {N - 1} \right)!}} \hfill \cr 
  y(N,c) = {{2N} \OVeR {\Left( {N + 2} \riGht)!}}c^{\,n + 2}  \hfill \cR 
  I(n - 1,c) = \SQrt n {D \oveR {Dc}}y(n,c) = {{2N\Sqrt n } \Over {\lEfT( {N + 1
}$, where $r$ is the radiu s of a $(n -2)$- sph ere ,lyin g in  the plane of$ T(n- 1,1)$ and centered wit h it; 
-  whi l e$r$ increas e sf r om$0 $toth e  i n-rad ius  $R_I =  1/\sqrt{n (n- 1) }$ the corre s po nding sphe reremains insi dethe si mp lex ,  with  asurfa ce are a  of
$$ 
S(n - 2)  = {{2\pi ^{\,\,( n -1)/2 } } \over {\Gamma \l e ft( {(n - 1)/2 } \rig ht ) }} r ^ {\, n -  2} 
$$
 -  a fter$ r$ hass ur p a s sed the in-radius , the spher e  it  indiv id uat e s will  part ly deb ord out ofthesimplex,
the c i rcle in red inthe sk etc h;thes ec ti onof its ar eai nte rceptedby  t he si mple x w i ll b e t hatinsid e the solid a ngl es d e fin ed by  thevert ic es;
- theinter ce pted area willbeco me  nullatth e c ir cum-r a dius $ R_C  =\sqrt{( n-1)/n} $ .
T he p r ob lem is that for hi gh $ n$ , the vo lume g e ts  c o ncentrat ed  on  the b order  oft he  sphere. 
 This ma ke s so th at  the m ax imu m o f $PD F (r)$  moves  beyondthe i n -radius and it becomes funda m en t a lt o ex pre ss
the area  int e rcep tedb ythe verti ces.Ic ou l d not yet succeed i nthat,and c ould not find  on the we b a  simpleform u la t ion.    
--- 2 nd st ep  ---
Le t 's summa rizeour know ledge til l here and  in tro duc e $ J ( n- 1,c)$ as thes u m of  t he 2ndmom ents of  $T (n- 1,c )$wr t its cen troid.
$ $\l ef t\ { \ matri x {
  d\le ft ( { n, c}\righ t ) = {c  \ove r {\ sq rt n } } \hfil l  \ c r  
 A\ le ft({n-1,c}\rig h t)= \sqrt  n {{c^{\ ,n-  1}}\o ver {\l eft( {n - 1}\r ight)!}} \ hf ill  \cr Y(n,c) =  {{2n} \over {\left( {n + 2} \r igh t)!}} c^{\ ,n + 2} \hf ill \c r  I(n -  1,c)= \sq rt  n{ d  \ove r {d c}} Y( n,c) = {{2 n \ sqr t n }  \ over  {\left ( {n + 1
}$, where_$r$ is_the radius of a_$(n-2)$-sphere, lying_in_the plane_of_$T(n-1,1)$ and centered_with it;
 -_while $r$  increases_from $0$ to_the_in-radius $R_I = 1/\sqrt{n(n-1)}$ the corresponding sphere remains inside the simplex, with a surface_area_of
$$
S(n -_2)_=_{{2\pi ^{\,\,(n - 1)/2} }_\over {\Gamma \left( {(n -_1)/2} \right)}}r^{\,n_- 2} 
$$
 - after $r$ has surpassed_the_in-radius, the sphere_it individuates will partly debord out of the simplex,_
the circle in red in the_sketch; the section_of_its_area intercepted by the_simplex will be that inside the_solid angles defined by the vertices;
_- the intercepted area will become _null at the circum-radius $R_C =_\sqrt{(n-1)/n}$.
The problem is that for_high $n$,_the volume gets concentrated on_the border of_the sphere.
_This makes so_that the maximum of $PDF(r)$ moves_beyond the in-radius_and it becomes fundamental to express
the_area_intercepted by the_vertices.
I_could_not yet_succeed in that,_and_could not_find_on the web a simple formulation.__  
--- 2nd step  ---
Let's_summarize our knowledge till_here_and introduce $J(n-1,c)$ as_the sum of the 2nd_moments of $T(n-1,c)$ wrt its centroid.
$$
\left\{_\matrix{
 _d\left( {n,c}_\right) = {c \over {\sqrt n }} \hfill \cr 
 _A\left( {n - 1,c} \right) =_\sqrt n {{c^{\,n -_1} }_\over_{\left( {n -_1}_\right)!}} \hfill_\cr 
  Y(n,c) = {{2n} \over_{\left( {n_+ 2} \right)!}}c^{\,n + 2} _\hfill \cr 
 _I(n_- 1,c) = \sqrt n {d_\over {dc}}Y(n,c) = {{2n\sqrt n }_\over {\left( {n + 1